%------------------------------------------------------------------------------
% File     : ITP235_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_InsertCorrectness 01015_065053
% 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    : 0067_VEBT_InsertCorrectness_01015_065053 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 11693 (3997 unt;1708 typ;   0 def)
%            Number of atoms       : 18384 (7013 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 21719 (2111   ~; 349   |;2271   &)
%                                         (2383 <=>;14605  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   38 (   2 avg)
%            Number of FOOLs       : 1231 ( 899 fml; 332 var)
%            Number of X terms     :  594 (   0  []; 521 ite;  73 let)
%            Number of types       :   14 (  13 usr)
%            Number of type conns  : 1546 (1271   >; 275   *;   0   +;   0  <<)
%            Number of predicates  :  291 ( 288 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1421 (1421 usr;  56 con; 0-8 aty)
%            Number of variables   : 35393 (31772   !; 949   ?;35393   :)
%                                         (2672  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 21:53:17.232
%------------------------------------------------------------------------------
% Could-be-implicit typings (21)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

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

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

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

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

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

tff(ty_t_String_Oliteral,type,
    literal: $tType ).

tff(ty_t_Sum__Type_Osum,type,
    sum_sum: ( $tType * $tType ) > $tType ).

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

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

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

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

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

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

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

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

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

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

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

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

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

% Explicit typings (1687)
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_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_Finite__Set_Ofinite,type,
    finite_finite: 
      !>[A: $tType] : $o ).

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

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

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

tff(sy_cl_Enum_Ofinite__lattice,type,
    finite_lattice: 
      !>[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_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_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_Obounded__lattice,type,
    bounded_lattice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

tff(sy_cl_Complete__Partial__Order_Occpo,type,
    comple9053668089753744459l_ccpo: 
      !>[A: $tType] : $o ).

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ot1__space,type,
    topological_t1_space: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Odiscrete__topology,type,
    topolo8865339358273720382pology: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__agn____,type,
    aTP_Lamp_agn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__ais____,type,
    aTP_Lamp_ais: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ait____,type,
    aTP_Lamp_ait: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aiu____,type,
    aTP_Lamp_aiu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiv____,type,
    aTP_Lamp_aiv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiw____,type,
    aTP_Lamp_aiw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aix____,type,
    aTP_Lamp_aix: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiy____,type,
    aTP_Lamp_aiy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiz____,type,
    aTP_Lamp_aiz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__aja____,type,
    aTP_Lamp_aja: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajb____,type,
    aTP_Lamp_ajb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajc____,type,
    aTP_Lamp_ajc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__aje____,type,
    aTP_Lamp_aje: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajf____,type,
    aTP_Lamp_ajf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajg____,type,
    aTP_Lamp_ajg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajh____,type,
    aTP_Lamp_ajh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__ajj____,type,
    aTP_Lamp_ajj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajk____,type,
    aTP_Lamp_ajk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajl____,type,
    aTP_Lamp_ajl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajm____,type,
    aTP_Lamp_ajm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajn____,type,
    aTP_Lamp_ajn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajo____,type,
    aTP_Lamp_ajo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajp____,type,
    aTP_Lamp_ajp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajq____,type,
    aTP_Lamp_ajq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajr____,type,
    aTP_Lamp_ajr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajs____,type,
    aTP_Lamp_ajs: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajt____,type,
    aTP_Lamp_ajt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aju____,type,
    aTP_Lamp_aju: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajv____,type,
    aTP_Lamp_ajv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajw____,type,
    aTP_Lamp_ajw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajx____,type,
    aTP_Lamp_ajx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajy____,type,
    aTP_Lamp_ajy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajz____,type,
    aTP_Lamp_ajz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__aka____,type,
    aTP_Lamp_aka: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akb____,type,
    aTP_Lamp_akb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akc____,type,
    aTP_Lamp_akc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akd____,type,
    aTP_Lamp_akd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ake____,type,
    aTP_Lamp_ake: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akf____,type,
    aTP_Lamp_akf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akg____,type,
    aTP_Lamp_akg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akh____,type,
    aTP_Lamp_akh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aki____,type,
    aTP_Lamp_aki: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akj____,type,
    aTP_Lamp_akj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akk____,type,
    aTP_Lamp_akk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akl____,type,
    aTP_Lamp_akl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akm____,type,
    aTP_Lamp_akm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akn____,type,
    aTP_Lamp_akn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__akp____,type,
    aTP_Lamp_akp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akq____,type,
    aTP_Lamp_akq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__akr____,type,
    aTP_Lamp_akr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aks____,type,
    aTP_Lamp_aks: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akt____,type,
    aTP_Lamp_akt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aku____,type,
    aTP_Lamp_aku: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akv____,type,
    aTP_Lamp_akv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akw____,type,
    aTP_Lamp_akw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akx____,type,
    aTP_Lamp_akx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aky____,type,
    aTP_Lamp_aky: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akz____,type,
    aTP_Lamp_akz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__ala____,type,
    aTP_Lamp_ala: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__gm____,type,
    aTP_Lamp_gm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gn____,type,
    aTP_Lamp_gn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__go____,type,
    aTP_Lamp_go: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__qn____,type,
    aTP_Lamp_qn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qo____,type,
    aTP_Lamp_qo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__vm____,type,
    aTP_Lamp_vm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vn____,type,
    aTP_Lamp_vn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vo____,type,
    aTP_Lamp_vo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vp____,type,
    aTP_Lamp_vp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
    bNF_Ca4139267488887388095finite: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
    bNF_Cardinal_csum: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) ) > set(product_prod(sum_sum(A,B),sum_sum(A,B))) ) ).

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

tff(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
    bNF_Ca8387033319878233205ardSuc: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
    bNF_Ca6860139660246222851ard_of: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
    bNF_Ca8970107618336181345der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

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

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

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

tff(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
    bNF_Ca7133664381575040944arCard: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

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

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

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

tff(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
    bNF_Greatest_image2p: 
      !>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( fun(C,A) * fun(D,B) * fun(C,fun(D,$o)) ) > fun(A,fun(B,$o)) ) ).

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

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

tff(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
    bNF_Gr1419584066657907630d_pred: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > $o ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Binomial_Obinomial,type,
    binomial: ( nat * nat ) > nat ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int * int ) > int ).

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : ( A > fun(nat,$o) ) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).

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

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: ( num * num ) > option(num) ).

tff(sy_c_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_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
    boolea5476839437570043046axioms: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * fun(A,fun(A,A)) ) > $o ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

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

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

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

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

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

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_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_Equiv__Relations_Ocongruent,type,
    equiv_congruent: 
      !>[A: $tType,B: $tType] : ( ( set(product_prod(A,A)) * fun(A,B) ) > $o ) ).

tff(sy_c_Equiv__Relations_Ocongruent2,type,
    equiv_congruent2: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,A)) * set(product_prod(B,B)) * fun(A,fun(B,C)) ) > $o ) ).

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

tff(sy_c_Equiv__Relations_Oproj,type,
    equiv_proj: 
      !>[B: $tType,A: $tType] : ( set(product_prod(B,A)) > fun(B,set(A)) ) ).

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

tff(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
    euclid6346220572633701492n_size: 
      !>[A: $tType] : ( A > nat ) ).

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

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

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

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

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

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__bot,type,
    at_bot: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Ocofinite,type,
    cofinite: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : fun(fun(A,$o),fun(filter(A),$o)) ).

tff(sy_c_Filter_Ofilter_OAbs__filter,type,
    abs_filter: 
      !>[A: $tType] : ( fun(fun(A,$o),$o) > filter(A) ) ).

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(fun(A,B),fun(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_Ofrequently,type,
    frequently: 
      !>[A: $tType] : fun(fun(A,$o),fun(filter(A),$o)) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( set(A) > fun(fun(A,B),fun(filter(A),filter(B))) ) ).

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

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

tff(sy_c_Filter_Orel__filter,type,
    rel_filter: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(filter(A),fun(filter(B),$o)) ) ).

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

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

tff(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
    finite673082921795544331dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
    finite4980608107308702382axioms: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

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

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

tff(sy_c_Finite__Set_Ofold__graph,type,
    finite_fold_graph: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > fun(B,$o) ) ).

tff(sy_c_Finite__Set_Ofolding,type,
    finite_folding: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__idem,type,
    finite_folding_idem: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__idem__axioms,type,
    finite7837460588564673216axioms: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,B)) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__idem__on,type,
    finite1890593828518410140dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
    finite6916993218817215295axioms: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on,type,
    finite_folding_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__on_OF,type,
    finite_folding_F: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B ) > fun(set(A),B) ) ).

tff(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( fun(B,C) > fun(fun(A,B),fun(A,C)) ) ).

tff(sy_c_Fun_Ofun__upd,type,
    fun_upd: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A * B ) > fun(A,B) ) ).

tff(sy_c_Fun_Oid,type,
    id: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Omap__fun,type,
    map_fun: 
      !>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( fun(C,A) * fun(B,D) * fun(A,B) ) > fun(C,D) ) ).

tff(sy_c_Fun_Ooverride__on,type,
    override_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * set(A) ) > fun(A,B) ) ).

tff(sy_c_Fun_Ostrict__mono__on,type,
    strict_mono_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).

tff(sy_c_Fun__Def_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_Fun__Def_Oreduction__pair,type,
    fun_reduction_pair: 
      !>[A: $tType] : ( product_prod(set(product_prod(A,A)),set(product_prod(A,A))) > $o ) ).

tff(sy_c_GCD_OGcd__class_OGcd,type,
    gcd_Gcd: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set,type,
    bounde6485984586167503788ce_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A * fun(A,A) ) > $o ) ).

tff(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
    bounde2362111253966948842tice_F: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * A ) > fun(set(A),A) ) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
    semiring_gcd_Gcd_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : ( 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] : ( 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(fun(B,A),fun(set(B),A)) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
    groups1962203154675924110t_prod: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : fun(fun(B,A),fun(A,fun(list(B),A))) ).

tff(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
    groups8242544230860333062m_list: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_HOL_ONO__MATCH,type,
    nO_MATCH: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_HOL_OUniq,type,
    uniq: 
      !>[A: $tType] : ( fun(A,$o) > $o ) ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

tff(sy_c_Hilbert__Choice_Oinv__into,type,
    hilbert_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > fun(B,A) ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
    complete_lattice_gfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
    complete_lattice_lfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
    infini527867602293511546merate: 
      !>[A: $tType] : ( set(A) > fun(nat,A) ) ).

tff(sy_c_Int_OAbs__Integ,type,
    abs_Integ: product_prod(nat,nat) > int ).

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: int > product_prod(nat,nat) ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Ointrel,type,
    intrel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Int_Onat,type,
    nat2: fun(int,nat) ).

tff(sy_c_Int_Opcr__int,type,
    pcr_int: fun(product_prod(nat,nat),fun(int,$o)) ).

tff(sy_c_Int_Opower__int,type,
    power_int: 
      !>[A: $tType] : ( ( A * int ) > A ) ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices_Oinf__class_Oinf,type,
    inf_inf: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices_Osup__class_Osup,type,
    sup_sup: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMin,type,
    lattic643756798350308766er_Min: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
    lattices_ord_arg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
    lattic7623131987881927897min_on: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > B ) ).

tff(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
    lattic501386751177426532rg_min: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,$o) ) > fun(B,$o) ) ).

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Osemilattice__order__set,type,
    lattic4895041142388067077er_set: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,$o)) * fun(A,fun(A,$o)) ) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__set,type,
    lattic149705377957585745ce_set: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > $o ) ).

tff(sy_c_Lattices__Big_Osemilattice__set_OF,type,
    lattic1715443433743089157tice_F: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Limits_OBfun,type,
    bfun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Limits_OZfun,type,
    zfun: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oarg__min__list,type,
    arg_min_list: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > A ) ).

tff(sy_c_List_Oarg__min__list__rel,type,
    arg_min_list_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),$o)) ).

tff(sy_c_List_Obutlast,type,
    butlast: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oconcat,type,
    concat: 
      !>[A: $tType] : ( list(list(A)) > list(A) ) ).

tff(sy_c_List_Ocoset,type,
    coset: 
      !>[A: $tType] : ( list(A) > set(A) ) ).

tff(sy_c_List_Ocount__list,type,
    count_list: 
      !>[A: $tType] : ( list(A) > fun(A,nat) ) ).

tff(sy_c_List_Odistinct,type,
    distinct: 
      !>[A: $tType] : ( list(A) > $o ) ).

tff(sy_c_List_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OdropWhile,type,
    dropWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(A) ) ).

tff(sy_c_List_Ofold,type,
    fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).

tff(sy_c_List_Ofolding__insort__key,type,
    folding_insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * set(B) * fun(B,A) ) > $o ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).

tff(sy_c_List_Ogen__length,type,
    gen_length: 
      !>[A: $tType] : ( nat > fun(list(A),nat) ) ).

tff(sy_c_List_Oinsert,type,
    insert: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_Olast,type,
    last: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexord,type,
    lexord: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olinorder_Oinsort__key,type,
    insort_key: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,A) * set(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
    linord329482645794927042rt_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Oinsort__key,type,
    linorder_insort_key: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * B * list(B) ) > list(B) ) ).

tff(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : ( set(A) > list(A) ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : ( A > fun(list(A),list(A)) ) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(list(A),fun(list(B),$o)) ) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( ( fun(A,Aa) * list(A) ) > list(Aa) ) ).

tff(sy_c_List_Olist_Orec__list,type,
    rec_list: 
      !>[C: $tType,A: $tType] : ( ( C * fun(A,fun(list(A),fun(C,C))) * list(A) ) > C ) ).

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_Olistrel,type,
    listrel: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(list(A),list(B))) ) ).

tff(sy_c_List_Olistrel1,type,
    listrel1: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olists,type,
    lists: 
      !>[A: $tType] : ( set(A) > set(list(A)) ) ).

tff(sy_c_List_Omeasures,type,
    measures: 
      !>[A: $tType] : ( list(fun(A,nat)) > set(product_prod(A,A)) ) ).

tff(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Onths,type,
    nths: 
      !>[A: $tType] : ( ( list(A) * set(nat) ) > list(A) ) ).

tff(sy_c_List_Onull,type,
    null: 
      !>[A: $tType] : ( list(A) > $o ) ).

tff(sy_c_List_Oord__class_Olexordp,type,
    ord_lexordp: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_Opartition,type,
    partition: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oremdups,type,
    remdups: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremdups__adj,type,
    remdups_adj: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oremove1,type,
    remove1: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_OremoveAll,type,
    removeAll: 
      !>[A: $tType] : ( ( A * list(A) ) > list(A) ) ).

tff(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Orev,type,
    rev: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Orotate,type,
    rotate: 
      !>[A: $tType] : ( nat > fun(list(A),list(A)) ) ).

tff(sy_c_List_Orotate1,type,
    rotate1: 
      !>[A: $tType] : ( list(A) > list(A) ) ).

tff(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).

tff(sy_c_List_Oshuffles,type,
    shuffles: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > set(list(A)) ) ).

tff(sy_c_List_Osorted__wrt,type,
    sorted_wrt: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) ) > $o ) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otake,type,
    take: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OtakeWhile,type,
    takeWhile: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Ounion,type,
    union: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_List_Ozip,type,
    zip: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_Map_Odom,type,
    dom: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(A) ) ).

tff(sy_c_Map_Ograph,type,
    graph: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(product_prod(A,B)) ) ).

tff(sy_c_Map_Omap__add,type,
    map_add: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(A,option(B)) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__of,type,
    map_of: 
      !>[A: $tType,B: $tType] : ( list(product_prod(A,B)) > fun(A,option(B)) ) ).

tff(sy_c_Map_Omap__upds,type,
    map_upds: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) * list(B) ) > fun(A,option(B)) ) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Map_Orestrict__map,type,
    restrict_map: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * set(A) ) > fun(A,option(B)) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Ocompow,type,
    compow: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Nat_Ofunpow,type,
    funpow: 
      !>[A: $tType] : fun(nat,fun(fun(A,A),fun(A,A))) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

tff(sy_c_Nat_Oold_Onat_Orec__nat,type,
    rec_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) ) > fun(nat,T) ) ).

tff(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
    rec_set_nat: 
      !>[T: $tType] : ( ( T * fun(nat,fun(T,T)) * nat ) > fun(T,$o) ) ).

tff(sy_c_Nat_Osemiring__1__class_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_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_Nat__Bijection_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_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: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: 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] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : ( option(A) > A ) ).

tff(sy_c_Option_Othese,type,
    these: 
      !>[A: $tType] : ( set(option(A)) > set(A) ) ).

tff(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
    order_532582986084564980_cclfp: 
      !>[A: $tType] : ( fun(A,A) > A ) ).

tff(sy_c_Order__Continuity_Oinf__continuous,type,
    order_inf_continuous: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Order__Continuity_Osup__continuous,type,
    order_sup_continuous: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Order__Relation_OAboveS,type,
    order_AboveS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > set(A) ) ).

tff(sy_c_Order__Relation_Olinear__order__on,type,
    order_679001287576687338der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Oofilter,type,
    order_ofilter: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * set(A) ) > $o ) ).

tff(sy_c_Order__Relation_Opartial__order__on,type,
    order_7125193373082350890der_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Opreorder__on,type,
    order_preorder_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Order__Relation_Ounder,type,
    order_under: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_OunderS,type,
    order_underS: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * A ) > set(A) ) ).

tff(sy_c_Order__Relation_Owell__order__on,type,
    order_well_order_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord_OLeast,type,
    least: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(fun(A,$o),A) ) ).

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oorder__class_Oantimono,type,
    order_antimono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oorder__class_Omono,type,
    order_mono: 
      !>[A: $tType,B: $tType] : fun(fun(A,B),$o) ).

tff(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
    order_strict_mono: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Orderings_Oordering__top,type,
    ordering_top: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_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] : ( 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_Oapfst,type,
    product_apfst: 
      !>[A: $tType,C: $tType,B: $tType] : ( fun(A,C) > fun(product_prod(A,B),product_prod(C,B)) ) ).

tff(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : ( fun(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(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: product_prod(int,int) > rat ).

tff(sy_c_Rat_OFract,type,
    fract: ( 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] : ( rat > A ) ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Opcr__rat,type,
    pcr_rat: fun(product_prod(int,int),fun(rat,$o)) ).

tff(sy_c_Rat_Opositive,type,
    positive: fun(rat,$o) ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Rat_Oratrel,type,
    ratrel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Real_OReal,type,
    real2: fun(nat,rat) > real ).

tff(sy_c_Real_Ocauchy,type,
    cauchy: fun(nat,rat) > $o ).

tff(sy_c_Real_Opcr__real,type,
    pcr_real: fun(fun(nat,rat),fun(real,$o)) ).

tff(sy_c_Real_Opositive,type,
    positive2: fun(real,$o) ).

tff(sy_c_Real_Orealrel,type,
    realrel: fun(fun(nat,rat),fun(fun(nat,rat),$o)) ).

tff(sy_c_Real_Orep__real,type,
    rep_real: fun(real,fun(nat,rat)) ).

tff(sy_c_Real_Ovanishes,type,
    vanishes: fun(nat,rat) > $o ).

tff(sy_c_Real__Vector__Spaces_OReals,type,
    real_Vector_Reals: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
    real_V2442710119149674383linear: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_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_Olinear,type,
    real_Vector_linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

tff(sy_c_Real__Vector__Spaces_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : ( real > A ) ).

tff(sy_c_Real__Vector__Spaces_Orepresentation,type,
    real_V7696804695334737415tation: 
      !>[A: $tType] : ( ( set(A) * A ) > 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_Real__Vector__Spaces_Osubspace,type,
    real_Vector_subspace: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Relation_ODomain,type,
    domain: 
      !>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(A)) ).

tff(sy_c_Relation_OField,type,
    field2: 
      !>[A: $tType] : fun(set(product_prod(A,A)),set(A)) ).

tff(sy_c_Relation_OId,type,
    id2: 
      !>[A: $tType] : set(product_prod(A,A)) ).

tff(sy_c_Relation_OId__on,type,
    id_on: 
      !>[A: $tType] : ( set(A) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_OImage,type,
    image: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > fun(set(A),set(B)) ) ).

tff(sy_c_Relation_ORange,type,
    range: 
      !>[A: $tType,B: $tType] : fun(set(product_prod(A,B)),set(B)) ).

tff(sy_c_Relation_ORangep,type,
    rangep: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > fun(B,$o) ) ).

tff(sy_c_Relation_Oantisym,type,
    antisym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oantisymp,type,
    antisymp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).

tff(sy_c_Relation_Oconverse,type,
    converse: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > set(product_prod(B,A)) ) ).

tff(sy_c_Relation_Oinv__image,type,
    inv_image: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Relation_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Oirreflp,type,
    irreflp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).

tff(sy_c_Relation_Orefl__on,type,
    refl_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Orelcomp,type,
    relcomp: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( set(product_prod(A,B)) * set(product_prod(B,C)) ) > set(product_prod(A,C)) ) ).

tff(sy_c_Relation_Osingle__valued,type,
    single_valued: 
      !>[A: $tType,B: $tType] : ( set(product_prod(A,B)) > $o ) ).

tff(sy_c_Relation_Osingle__valuedp,type,
    single_valuedp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Relation_Osym,type,
    sym: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Ototal__on,type,
    total_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

tff(sy_c_Relation_Otrans,type,
    trans: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Otransp,type,
    transp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > $o ) ).

tff(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
    algebr8660921524188924756oprime: 
      !>[A: $tType] : ( ( A * A ) > $o ) ).

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun($o,A) ).

tff(sy_c_Series_Osuminf,type,
    suminf: 
      !>[A: $tType] : ( fun(nat,A) > A ) ).

tff(sy_c_Series_Osummable,type,
    summable: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( ( fun(nat,A) * A ) > $o ) ).

tff(sy_c_Set_OBall,type,
    ball: 
      !>[A: $tType] : ( set(A) > fun(fun(A,$o),$o) ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : fun(fun(A,$o),set(A)) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

tff(sy_c_Set_Odisjnt,type,
    disjnt: 
      !>[A: $tType] : fun(set(A),fun(set(A),$o)) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image2: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert2: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Ois__empty,type,
    is_empty: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Ois__singleton,type,
    is_singleton: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Set_Opairwise,type,
    pairwise: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Set_Oremove,type,
    remove: 
      !>[A: $tType] : fun(A,fun(set(A),set(A))) ).

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set_Ovimage,type,
    vimage: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(B) ) > set(A) ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

tff(sy_c_Set__Interval_Oord__class_OatLeast,type,
    set_ord_atLeast: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
    set_ord_greaterThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : fun(A,set(A)) ).

tff(sy_c_String_OCode_Oabort,type,
    abort: 
      !>[A: $tType] : ( ( literal * fun(product_unit,A) ) > A ) ).

tff(sy_c_String_OLiteral,type,
    literal2: ( $o * $o * $o * $o * $o * $o * $o * literal ) > literal ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

tff(sy_c_Sum__Type_OPlus,type,
    sum_Plus: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(sum_sum(A,B)) ) ).

tff(sy_c_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: real > real ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( fun(nat,A) > fun(nat,A) ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Opowr__real,type,
    powr_real: ( real * real ) > real ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transfer_Obi__total,type,
    bi_total: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $o ) ).

tff(sy_c_Transfer_Obi__unique,type,
    bi_unique: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,$o)) > $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: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: ( vEBT_VEBT * extended_enat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: fun(product_prod(vEBT_VEBT,extended_enat),fun(product_prod(vEBT_VEBT,extended_enat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(A,$o) ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

tff(sy_c_Wellfounded_Oless__than,type,
    less_than: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omax__extp,type,
    max_extp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * set(A) * set(A) ) > $o ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set(product_prod(nat,nat)) ).

tff(sy_c_Wellfounded_Owf,type,
    wf: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Zorn_OChains,type,
    chains: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(set(A)) ) ).

tff(sy_c_Zorn_Ochain__subset,type,
    chain_subset: 
      !>[A: $tType] : ( set(set(A)) > $o ) ).

tff(sy_c_Zorn_Ochains,type,
    chains2: 
      !>[A: $tType] : ( set(set(A)) > set(set(set(A))) ) ).

tff(sy_c_Zorn_Oinit__seg__of,type,
    init_seg_of: 
      !>[A: $tType] : set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))) ).

tff(sy_c_Zorn_Opred__on_Ochain,type,
    pred_chain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) ) > fun(set(A),$o) ) ).

tff(sy_c_Zorn_Opred__on_Omaxchain,type,
    pred_maxchain: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) * set(A) ) > $o ) ).

tff(sy_c_Zorn_Opred__on_Osuc,type,
    pred_suc: 
      !>[A: $tType] : ( ( set(A) * fun(A,fun(A,$o)) * set(A) ) > set(A) ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fAll,type,
    fAll: 
      !>[A: $tType] : fun(fun(A,$o),$o) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( ( A * set(A) ) > $o ) ).

tff(sy_v_n,type,
    n: nat ).

tff(sy_v_t,type,
    t: vEBT_VEBT ).

tff(sy_v_x,type,
    x: nat ).

tff(sy_v_y____,type,
    y: nat ).

% Relevant facts (9148)
tff(fact_0__092_060open_062y_A_092_060in_062_Aset__vebt_H_A_Ivebt__insert_At_Ax_J_092_060close_062,axiom,
    member(nat,y,vEBT_VEBT_set_vebt(vEBT_vebt_insert(t,x))) ).

% \<open>y \<in> set_vebt' (vebt_insert t x)\<close>
tff(fact_1_buildup__gives__empty,axiom,
    ! [Na: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Na)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_2_Un__insert__left,axiom,
    ! [A: $tType,A3: A,B2: set(A),C2: 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(A,fun(set(A),set(A)),insert2(A),A3),B2)),C2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2)) ).

% Un_insert_left
tff(fact_3_Un__insert__right,axiom,
    ! [A: $tType,A4: set(A),A3: A,B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ).

% Un_insert_right
tff(fact_4_Un__empty,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = bot_bot(set(A)) )
    <=> ( ( A4 = bot_bot(set(A)) )
        & ( B2 = bot_bot(set(A)) ) ) ) ).

% Un_empty
tff(fact_5_singletonI,axiom,
    ! [A: $tType,A3: A] : member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ).

% singletonI
tff(fact_6_sup__bot__left,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),Xa) = Xa ) ).

% sup_bot_left
tff(fact_7_sup__bot__right,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ).

% sup_bot_right
tff(fact_8_bot__eq__sup__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Ya: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Ya = bot_bot(A) ) ) ) ) ).

% bot_eq_sup_iff
tff(fact_9_sup__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) = bot_bot(A) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Ya = bot_bot(A) ) ) ) ) ).

% sup_eq_bot_iff
tff(fact_10_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = bot_bot(A) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B3 = bot_bot(A) ) ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_11_sup__bot_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),bot_bot(A)),A3) = A3 ) ).

% sup_bot.left_neutral
tff(fact_12_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(A) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
        <=> ( ( A3 = bot_bot(A) )
            & ( B3 = bot_bot(A) ) ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_13_empty__Collect__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),P) )
    <=> ! [X: A] : ~ aa(A,$o,P,X) ) ).

% empty_Collect_eq
tff(fact_14_Collect__empty__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ! [X: A] : ~ aa(A,$o,P,X) ) ).

% Collect_empty_eq
tff(fact_15_all__not__in__conv,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ! [X: A] : ~ member(A,X,A4)
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% all_not_in_conv
tff(fact_16_empty__iff,axiom,
    ! [A: $tType,C3: A] : ~ member(A,C3,bot_bot(set(A))) ).

% empty_iff
tff(fact_17_insert__absorb2,axiom,
    ! [A: $tType,Xa: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4) ).

% insert_absorb2
tff(fact_18_insert__iff,axiom,
    ! [A: $tType,A3: A,B3: A,A4: set(A)] :
      ( member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),A4))
    <=> ( ( A3 = B3 )
        | member(A,A3,A4) ) ) ).

% insert_iff
tff(fact_19_insertCI,axiom,
    ! [A: $tType,A3: A,B2: set(A),B3: A] :
      ( ( ~ member(A,A3,B2)
       => ( A3 = B3 ) )
     => member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B2)) ) ).

% insertCI
tff(fact_20_sup__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xa: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),sup_sup(fun(B,A)),F2),G),Xa) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,Xa)),aa(B,A,G,Xa)) ) ).

% sup_apply
tff(fact_21_sup_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ).

% sup.right_idem
tff(fact_22_sup__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) ) ).

% sup_left_idem
tff(fact_23_sup_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ).

% sup.left_idem
tff(fact_24_sup__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Xa) = Xa ) ).

% sup_idem
tff(fact_25_sup_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),A3) = A3 ) ).

% sup.idem
tff(fact_26_Un__iff,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
    <=> ( member(A,C3,A4)
        | member(A,C3,B2) ) ) ).

% Un_iff
tff(fact_27_UnCI,axiom,
    ! [A: $tType,C3: A,B2: set(A),A4: set(A)] :
      ( ( ~ member(A,C3,B2)
       => member(A,C3,A4) )
     => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ) ).

% UnCI
tff(fact_28_sup__bot_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),bot_bot(A)) = A3 ) ).

% sup_bot.right_neutral
tff(fact_29_bot__set__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) ).

% bot_set_def
tff(fact_30_ex__in__conv,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ? [X: A] : member(A,X,A4)
    <=> ( A4 != bot_bot(set(A)) ) ) ).

% ex_in_conv
tff(fact_31_equals0I,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ! [Y: A] : ~ member(A,Y,A4)
     => ( A4 = bot_bot(set(A)) ) ) ).

% equals0I
tff(fact_32_equals0D,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( ( A4 = bot_bot(set(A)) )
     => ~ member(A,A3,A4) ) ).

% equals0D
tff(fact_33_emptyE,axiom,
    ! [A: $tType,A3: A] : ~ member(A,A3,bot_bot(set(A))) ).

% emptyE
tff(fact_34_mk__disjoint__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( member(A,A3,A4)
     => ? [B4: set(A)] :
          ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B4) )
          & ~ member(A,A3,B4) ) ) ).

% mk_disjoint_insert
tff(fact_35_insert__commute,axiom,
    ! [A: $tType,Xa: A,Ya: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),A4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) ).

% insert_commute
tff(fact_36_insert__eq__iff,axiom,
    ! [A: $tType,A3: A,A4: set(A),B3: A,B2: set(A)] :
      ( ~ member(A,A3,A4)
     => ( ~ member(A,B3,B2)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B2) )
        <=> $ite(
              A3 = B3,
              A4 = B2,
              ? [C4: set(A)] :
                ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),C4) )
                & ~ member(A,B3,C4)
                & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),C4) )
                & ~ member(A,A3,C4) ) ) ) ) ) ).

% insert_eq_iff
tff(fact_37_insert__absorb,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( member(A,A3,A4)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = A4 ) ) ).

% insert_absorb
tff(fact_38_insert__ident,axiom,
    ! [A: $tType,Xa: A,A4: set(A),B2: set(A)] :
      ( ~ member(A,Xa,A4)
     => ( ~ member(A,Xa,B2)
       => ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),B2) )
        <=> ( A4 = B2 ) ) ) ) ).

% insert_ident
tff(fact_39_Set_Oset__insert,axiom,
    ! [A: $tType,Xa: A,A4: set(A)] :
      ( member(A,Xa,A4)
     => ~ ! [B4: set(A)] :
            ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),B4) )
           => member(A,Xa,B4) ) ) ).

% Set.set_insert
tff(fact_40_insertI2,axiom,
    ! [A: $tType,A3: A,B2: set(A),B3: A] :
      ( member(A,A3,B2)
     => member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B2)) ) ).

% insertI2
tff(fact_41_insertI1,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) ).

% insertI1
tff(fact_42_insertE,axiom,
    ! [A: $tType,A3: A,B3: A,A4: set(A)] :
      ( member(A,A3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),A4))
     => ( ( A3 != B3 )
       => member(A,A3,A4) ) ) ).

% insertE
tff(fact_43_sup__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_sup(B)
     => ! [F2: fun(A,B),G: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),sup_sup(fun(A,B)),F2),G),X2) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) ) ).

% sup_fun_def
tff(fact_44_sup__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ).

% sup_left_commute
tff(fact_45_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      ( member(A,A3,aa(fun(A,$o),set(A),collect(A),P))
    <=> aa(A,$o,P,A3) ) ).

% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
    ! [A: $tType,A4: set(A)] : aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)) = A4 ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
        <=> aa(A,$o,Q,X3) )
     => ( aa(fun(A,$o),set(A),collect(A),P) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).

% Collect_cong
tff(fact_48_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_49_sup_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C3)) ) ).

% sup.left_commute
tff(fact_50_sup__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Xa) ) ).

% sup_commute
tff(fact_51_sup_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),A3) ) ).

% sup.commute
tff(fact_52_sup__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)) ) ).

% sup_assoc
tff(fact_53_sup_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C3)) ) ).

% sup.assoc
tff(fact_54_inf__sup__aci_I5_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Xa) ) ).

% inf_sup_aci(5)
tff(fact_55_inf__sup__aci_I6_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)),Z) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)) ) ).

% inf_sup_aci(6)
tff(fact_56_inf__sup__aci_I7_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ).

% inf_sup_aci(7)
tff(fact_57_inf__sup__aci_I8_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) ) ).

% inf_sup_aci(8)
tff(fact_58_Un__left__commute,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C2)) ).

% Un_left_commute
tff(fact_59_Un__left__absorb,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) ).

% Un_left_absorb
tff(fact_60_Un__commute,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),A4) ).

% Un_commute
tff(fact_61_Un__absorb,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),A4) = A4 ).

% Un_absorb
tff(fact_62_Un__assoc,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: 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)),A4),B2)),C2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2)) ).

% Un_assoc
tff(fact_63_ball__Un,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
         => aa(A,$o,P,X) )
    <=> ( ! [X: A] :
            ( member(A,X,A4)
           => aa(A,$o,P,X) )
        & ! [X: A] :
            ( member(A,X,B2)
           => aa(A,$o,P,X) ) ) ) ).

% ball_Un
tff(fact_64_bex__Un,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
          & aa(A,$o,P,X) )
    <=> ( ? [X: A] :
            ( member(A,X,A4)
            & aa(A,$o,P,X) )
        | ? [X: A] :
            ( member(A,X,B2)
            & aa(A,$o,P,X) ) ) ) ).

% bex_Un
tff(fact_65_UnI2,axiom,
    ! [A: $tType,C3: A,B2: set(A),A4: set(A)] :
      ( member(A,C3,B2)
     => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ) ).

% UnI2
tff(fact_66_UnI1,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,A4)
     => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ) ).

% UnI1
tff(fact_67_UnE,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
     => ( ~ member(A,C3,A4)
       => member(A,C3,B2) ) ) ).

% UnE
tff(fact_68_singleton__inject,axiom,
    ! [A: $tType,A3: A,B3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))) )
     => ( A3 = B3 ) ) ).

% singleton_inject
tff(fact_69_insert__not__empty,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) != bot_bot(set(A)) ).

% insert_not_empty
tff(fact_70_doubleton__eq__iff,axiom,
    ! [A: $tType,A3: A,B3: A,C3: A,D2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),D2),bot_bot(set(A)))) )
    <=> ( ( ( A3 = C3 )
          & ( B3 = D2 ) )
        | ( ( A3 = D2 )
          & ( B3 = C3 ) ) ) ) ).

% doubleton_eq_iff
tff(fact_71_singleton__iff,axiom,
    ! [A: $tType,B3: A,A3: A] :
      ( member(A,B3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))
    <=> ( B3 = A3 ) ) ).

% singleton_iff
tff(fact_72_singletonD,axiom,
    ! [A: $tType,B3: A,A3: A] :
      ( member(A,B3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))
     => ( B3 = A3 ) ) ).

% singletonD
tff(fact_73_Un__empty__right,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),bot_bot(set(A))) = A4 ).

% Un_empty_right
tff(fact_74_Un__empty__left,axiom,
    ! [A: $tType,B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),bot_bot(set(A))),B2) = B2 ).

% Un_empty_left
tff(fact_75_singleton__Un__iff,axiom,
    ! [A: $tType,Xa: A,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) )
    <=> ( ( ( A4 = bot_bot(set(A)) )
          & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ) )
        | ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) )
          & ( B2 = bot_bot(set(A)) ) )
        | ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) )
          & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% singleton_Un_iff
tff(fact_76_Un__singleton__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),Xa: A] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) )
    <=> ( ( ( A4 = bot_bot(set(A)) )
          & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ) )
        | ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) )
          & ( B2 = bot_bot(set(A)) ) )
        | ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) )
          & ( B2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ) ) ) ) ).

% Un_singleton_iff
tff(fact_77_insert__is__Un,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))),A4) ).

% insert_is_Un
tff(fact_78__092_060open_062set__vebt_H_At_A_092_060union_062_A_123x_125_A_092_060subseteq_062_Aset__vebt_H_A_Ivebt__insert_At_Ax_J_092_060close_062,axiom,
    aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_VEBT_set_vebt(t)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),x),bot_bot(set(nat))))),vEBT_VEBT_set_vebt(vEBT_vebt_insert(t,x))) ).

% \<open>set_vebt' t \<union> {x} \<subseteq> set_vebt' (vebt_insert t x)\<close>
tff(fact_79_buildup__nothing__in__leaf,axiom,
    ! [Na: nat,Xa: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Na),Xa) ).

% buildup_nothing_in_leaf
tff(fact_80_assms_I1_J,axiom,
    vEBT_invar_vebt(t,n) ).

% assms(1)
tff(fact_81_the__elem__eq,axiom,
    ! [A: $tType,Xa: A] : the_elem(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ).

% the_elem_eq
tff(fact_82_bot__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( bot(A)
     => ! [Xa: B] : aa(B,A,bot_bot(fun(B,A)),Xa) = bot_bot(A) ) ).

% bot_apply
tff(fact_83_buildup__nothing__in__min__max,axiom,
    ! [Na: nat,Xa: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Na),Xa) ).

% buildup_nothing_in_min_max
tff(fact_84_is__singletonI,axiom,
    ! [A: $tType,Xa: A] : is_singleton(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ).

% is_singletonI
tff(fact_85_boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),bot_bot(A)) = Xa ) ).

% boolean_algebra.disj_zero_right
tff(fact_86_Set_Ois__empty__def,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_empty(A,A4)
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% Set.is_empty_def
tff(fact_87_is__singleton__def,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
    <=> ? [X: A] : A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ).

% is_singleton_def
tff(fact_88_is__singletonE,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
     => ~ ! [X3: A] : A4 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))) ) ).

% is_singletonE
tff(fact_89_boolean__algebra__cancel_Osup2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,K: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),B3) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B2) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ) ).

% boolean_algebra_cancel.sup2
tff(fact_90_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),A3) ) ).

% dual_order.refl
tff(fact_91_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Xa) ) ).

% order_refl
tff(fact_92_subsetI,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ! [X3: A] :
          ( member(A,X3,A4)
         => member(A,X3,B2) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ).

% subsetI
tff(fact_93_subset__antisym,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
       => ( A4 = B2 ) ) ) ).

% subset_antisym
tff(fact_94_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_95_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z) ) ) ) ).

% le_sup_iff
tff(fact_96_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C3)),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3) ) ) ) ).

% sup.bounded_iff
tff(fact_97_subset__empty,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),bot_bot(set(A)))
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_98_empty__subsetI,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A4) ).

% empty_subsetI
tff(fact_99_insert__subset,axiom,
    ! [A: $tType,Xa: A,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),B2)
    <=> ( member(A,Xa,B2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ) ).

% insert_subset
tff(fact_100_Un__subset__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)),C2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2) ) ) ).

% Un_subset_iff
tff(fact_101_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B3: A,A3: A,A4: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) )
    <=> ( ( A3 = B3 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_102_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A3: A,A4: set(A),B3: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))) )
    <=> ( ( A3 = B3 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_103_in__mono,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( member(A,Xa,A4)
       => member(A,Xa,B2) ) ) ).

% in_mono
tff(fact_104_subsetD,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( member(A,C3,A4)
       => member(A,C3,B2) ) ) ).

% subsetD
tff(fact_105_equalityE,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( A4 = B2 )
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4) ) ) ).

% equalityE
tff(fact_106_subset__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
    <=> ! [X: A] :
          ( member(A,X,A4)
         => member(A,X,B2) ) ) ).

% subset_eq
tff(fact_107_equalityD1,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( A4 = B2 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ).

% equalityD1
tff(fact_108_equalityD2,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( A4 = B2 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4) ) ).

% equalityD2
tff(fact_109_subset__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
    <=> ! [T2: A] :
          ( member(A,T2,A4)
         => member(A,T2,B2) ) ) ).

% subset_iff
tff(fact_110_subset__refl,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),A4) ).

% subset_refl
tff(fact_111_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
         => aa(A,$o,Q,X3) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ) ).

% Collect_mono
tff(fact_112_subset__trans,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2) ) ) ).

% subset_trans
tff(fact_113_set__eq__subset,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( A4 = B2 )
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4) ) ) ).

% set_eq_subset
tff(fact_114_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q))
    <=> ! [X: A] :
          ( aa(A,$o,P,X)
         => aa(A,$o,Q,X) ) ) ).

% Collect_mono_iff
tff(fact_115_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
          <=> ( Xa = Ya ) ) ) ) ).

% order_antisym_conv
tff(fact_116_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ).

% linorder_le_cases
tff(fact_117_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( ( aa(A,B,F2,B3) = C3 )
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),C3) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_118_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C3: B] :
          ( ( A3 = aa(B,A,F2,B3) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,C3)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_119_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ).

% linorder_linear
tff(fact_120_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa = Ya )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ).

% order_eq_refl
tff(fact_121_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B3)),C3)
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),C3) ) ) ) ) ).

% order_subst2
tff(fact_122_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,B3))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,C3)) ) ) ) ) ).

% order_subst1
tff(fact_123_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_124_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
        <=> ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,G,X)) ) ) ).

% le_fun_def
tff(fact_125_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G) ) ) ).

% le_funI
tff(fact_126_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,G,Xa)) ) ) ).

% le_funE
tff(fact_127_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B),Xa: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,G,Xa)) ) ) ).

% le_funD
tff(fact_128_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
           => ( A3 = B3 ) ) ) ) ).

% antisym
tff(fact_129_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3) ) ) ) ).

% dual_order.trans
tff(fact_130_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => ( A3 = B3 ) ) ) ) ).

% dual_order.antisym
tff(fact_131_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% dual_order.eq_iff
tff(fact_132_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B3: A] :
          ( ! [A5: A,B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B5)
             => aa(A,$o,aa(A,fun(A,$o),P,A5),B5) )
         => ( ! [A5: A,B5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B5),A5)
               => aa(A,$o,aa(A,fun(A,$o),P,A5),B5) )
           => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) ) ) ) ).

% linorder_wlog
tff(fact_133_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) ) ) ) ).

% order_trans
tff(fact_134_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3) ) ) ) ).

% order.trans
tff(fact_135_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
           => ( Xa = Ya ) ) ) ) ).

% order_antisym
tff(fact_136_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( ( B3 = C3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3) ) ) ) ).

% ord_le_eq_trans
tff(fact_137_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3) ) ) ) ).

% ord_eq_le_trans
tff(fact_138_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa = Ya )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ) ).

% order_class.order_eq_iff
tff(fact_139_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Ya) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Ya)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_140_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & ( B3 != A3 ) ) ) ) ).

% nle_le
tff(fact_141_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A3) ) ).

% bot.extremum
tff(fact_142_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),bot_bot(A))
        <=> ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_143_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),bot_bot(A))
         => ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_144_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Ya: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) ) ).

% inf_sup_ord(4)
tff(fact_145_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) ) ).

% inf_sup_ord(3)
tff(fact_146_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),Xa)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xa)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),Xa) ) ) ) ).

% le_supE
tff(fact_147_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,Xa: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),Xa) ) ) ) ).

% le_supI
tff(fact_148_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) ) ).

% sup_ge1
tff(fact_149_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ya: A,Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) ) ).

% sup_ge2
tff(fact_150_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% le_supI1
tff(fact_151_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% le_supI2
tff(fact_152_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A3: A,D2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ) ).

% sup.mono
tff(fact_153_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,C3: A,B3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C3),D2)) ) ) ) ).

% sup_mono
tff(fact_154_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ya: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)),Xa) ) ) ) ).

% sup_least
tff(fact_155_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) = Ya ) ) ) ).

% le_iff_sup
tff(fact_156_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ) ).

% sup.orderE
tff(fact_157_sup_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% sup.orderI
tff(fact_158_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F2: fun(A,fun(A,A)),Xa: A,Ya: A] :
          ( ! [X3: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F2,X3),Y))
         => ( ! [X3: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),F2,X3),Y))
           => ( ! [X3: A,Y: A,Z2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),X3)
                   => aa(A,$o,aa(A,fun(A,$o),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),Xa),Ya) = aa(A,A,aa(A,fun(A,A),F2,Xa),Ya) ) ) ) ) ) ).

% sup_unique
tff(fact_159_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb1
tff(fact_160_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb2
tff(fact_161_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) = Xa ) ) ) ).

% sup_absorb1
tff(fact_162_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) = Ya ) ) ) ).

% sup_absorb2
tff(fact_163_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C3)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3) ) ) ) ).

% sup.boundedE
tff(fact_164_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C3)),A3) ) ) ) ).

% sup.boundedI
tff(fact_165_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ) ) ).

% sup.order_iff
tff(fact_166_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ).

% sup.cobounded1
tff(fact_167_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ).

% sup.cobounded2
tff(fact_168_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb_iff1
tff(fact_169_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb_iff2
tff(fact_170_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.coboundedI1
tff(fact_171_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.coboundedI2
tff(fact_172_insert__mono,axiom,
    ! [A: $tType,C2: set(A),D3: set(A),A3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),D3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),C2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),D3)) ) ).

% insert_mono
tff(fact_173_subset__insert,axiom,
    ! [A: $tType,Xa: A,A4: set(A),B2: set(A)] :
      ( ~ member(A,Xa,A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),B2))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ) ).

% subset_insert
tff(fact_174_subset__insertI,axiom,
    ! [A: $tType,B2: set(A),A3: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) ).

% subset_insertI
tff(fact_175_subset__insertI2,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),B3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B2)) ) ).

% subset_insertI2
tff(fact_176_Un__mono,axiom,
    ! [A: $tType,A4: set(A),C2: set(A),B2: set(A),D3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),D3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C2),D3)) ) ) ).

% Un_mono
tff(fact_177_Un__least,axiom,
    ! [A: $tType,A4: set(A),C2: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)),C2) ) ) ).

% Un_least
tff(fact_178_Un__upper1,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ).

% Un_upper1
tff(fact_179_Un__upper2,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ).

% Un_upper2
tff(fact_180_Un__absorb1,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = B2 ) ) ).

% Un_absorb1
tff(fact_181_Un__absorb2,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = A4 ) ) ).

% Un_absorb2
tff(fact_182_subset__UnE,axiom,
    ! [A: $tType,C2: set(A),A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
     => ~ ! [A6: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),A4)
           => ! [B6: set(A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B6),B2)
               => ( C2 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A6),B6) ) ) ) ) ).

% subset_UnE
tff(fact_183_subset__Un__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = B2 ) ) ).

% subset_Un_eq
tff(fact_184_subset__singletonD,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))
     => ( ( A4 = bot_bot(set(A)) )
        | ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_185_subset__singleton__iff,axiom,
    ! [A: $tType,X4: set(A),A3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))
    <=> ( ( X4 = bot_bot(set(A)) )
        | ( X4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_186_is__singleton__the__elem,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
    <=> ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the_elem(A,A4)),bot_bot(set(A))) ) ) ).

% is_singleton_the_elem
tff(fact_187_is__singletonI_H,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ! [X3: A,Y: A] :
            ( member(A,X3,A4)
           => ( member(A,Y,A4)
             => ( X3 = Y ) ) )
       => is_singleton(A,A4) ) ) ).

% is_singletonI'
tff(fact_188_bot__fun__def,axiom,
    ! [A: $tType,B: $tType] :
      ( bot(B)
     => ! [X2: A] : aa(A,B,bot_bot(fun(A,B)),X2) = bot_bot(B) ) ).

% bot_fun_def
tff(fact_189_boolean__algebra__cancel_Osup1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: A,K: A,A3: A,B3: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),K),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ) ).

% boolean_algebra_cancel.sup1
tff(fact_190_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(Tree,Na)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xa)
       => ( vEBT_V5719532721284313246member(Tree,Xa)
          | vEBT_VEBT_membermima(Tree,Xa) ) ) ) ).

% member_valid_both_member_options
tff(fact_191_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
     => vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq2
tff(fact_192_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(Ta,D2)
     => vEBT_VEBT_valid(Ta,D2) ) ).

% valid_eq1
tff(fact_193_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
    <=> vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq
tff(fact_194_bot__empty__eq,axiom,
    ! [A: $tType,X2: A] :
      ( aa(A,$o,bot_bot(fun(A,$o)),X2)
    <=> member(A,X2,bot_bot(set(A))) ) ).

% bot_empty_eq
tff(fact_195_Collect__empty__eq__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( aa(fun(A,$o),set(A),collect(A),P) = bot_bot(set(A)) )
    <=> ( P = bot_bot(fun(A,$o)) ) ) ).

% Collect_empty_eq_bot
tff(fact_196_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => aa(set(nat),$o,finite_finite2(nat),vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_197_insert__subsetI,axiom,
    ! [A: $tType,Xa: A,A4: set(A),X4: set(A)] :
      ( member(A,Xa,A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),A4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),X4)),A4) ) ) ).

% insert_subsetI
tff(fact_198_subset__emptyI,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ! [X3: A] : ~ member(A,X3,A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_199_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa)
    <=> ( vEBT_V5719532721284313246member(Ta,Xa)
        | vEBT_VEBT_membermima(Ta,Xa) ) ) ).

% both_member_options_def
tff(fact_200_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_201_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_202_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa)
      <=> aa(nat,$o,vEBT_vebt_member(Ta),Xa) ) ) ).

% both_member_options_equiv_member
tff(fact_203_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa)
       => aa(nat,$o,vEBT_vebt_member(Ta),Xa) ) ) ).

% valid_member_both_member_options
tff(fact_204_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( aa(nat,$o,vEBT_vebt_member(Ta),Xa)
      <=> member(nat,Xa,vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_205_finite__Un,axiom,
    ! [A: $tType,F3: set(A),G2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F3),G2))
    <=> ( aa(set(A),$o,finite_finite2(A),F3)
        & aa(set(A),$o,finite_finite2(A),G2) ) ) ).

% finite_Un
tff(fact_206_finite__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))
    <=> aa(set(A),$o,finite_finite2(A),A4) ) ).

% finite_insert
tff(fact_207_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),zero_zero(A))
        <=> ( Na = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_208_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_minNull(Ta)
     => ~ aa(nat,$o,vEBT_vebt_member(Ta),Xa) ) ).

% min_Null_member
tff(fact_209_finite__subset__induct_H,axiom,
    ! [A: $tType,F3: set(A),A4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F3),A4)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A5: A,F4: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F4)
               => ( member(A,A5,A4)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F4),A4)
                   => ( ~ member(A,A5,F4)
                     => ( aa(set(A),$o,P,F4)
                       => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),F4)) ) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_subset_induct'
tff(fact_210_finite__subset__induct,axiom,
    ! [A: $tType,F3: set(A),A4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F3),A4)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A5: A,F4: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F4)
               => ( member(A,A5,A4)
                 => ( ~ member(A,A5,F4)
                   => ( aa(set(A),$o,P,F4)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),F4)) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_subset_induct
tff(fact_211_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),P: fun(set(A),$o),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X3: A,S2: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),S2)
                 => ( ! [Y2: A] :
                        ( member(A,Y2,S2)
                       => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y2)),aa(A,B,F2,X3)) )
                   => ( aa(set(A),$o,P,S2)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),S2)) ) ) )
             => aa(set(A),$o,P,S) ) ) ) ) ).

% finite_ranking_induct
tff(fact_212_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Ta)
     => ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_1) ) ).

% not_min_Null_member
tff(fact_213_infinite__finite__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),A4: set(A)] :
      ( ! [A7: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A7)
         => aa(set(A),$o,P,A7) )
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X3: A,F4: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F4)
             => ( ~ member(A,X3,F4)
               => ( aa(set(A),$o,P,F4)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F4)) ) ) )
         => aa(set(A),$o,P,A4) ) ) ) ).

% infinite_finite_induct
tff(fact_214_finite__ne__induct,axiom,
    ! [A: $tType,F3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( ( F3 != bot_bot(set(A)) )
       => ( ! [X3: A] : aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))
         => ( ! [X3: A,F4: set(A)] :
                ( aa(set(A),$o,finite_finite2(A),F4)
               => ( ( F4 != bot_bot(set(A)) )
                 => ( ~ member(A,X3,F4)
                   => ( aa(set(A),$o,P,F4)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F4)) ) ) ) )
           => aa(set(A),$o,P,F3) ) ) ) ) ).

% finite_ne_induct
tff(fact_215_finite__induct,axiom,
    ! [A: $tType,F3: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [X3: A,F4: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),F4)
             => ( ~ member(A,X3,F4)
               => ( aa(set(A),$o,P,F4)
                 => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),F4)) ) ) )
         => aa(set(A),$o,P,F3) ) ) ) ).

% finite_induct
tff(fact_216_finite__code,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A4: set(A)] : aa(set(A),$o,finite_finite2(A),A4) ) ).

% finite_code
tff(fact_217_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: A] :
          ( ( zero_zero(A) = Xa )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_218_finite__set__choice,axiom,
    ! [B: $tType,A: $tType,A4: set(A),P: fun(A,fun(B,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X3: A] :
            ( member(A,X3,A4)
           => ? [X_12: B] : aa(B,$o,aa(A,fun(B,$o),P,X3),X_12) )
       => ? [F5: fun(A,B)] :
          ! [X2: A] :
            ( member(A,X2,A4)
           => aa(B,$o,aa(A,fun(B,$o),P,X2),aa(A,B,F5,X2)) ) ) ) ).

% finite_set_choice
tff(fact_219_finite,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => ! [A4: set(A)] : aa(set(A),$o,finite_finite2(A),A4) ) ).

% finite
tff(fact_220_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ).

% zero_le
tff(fact_221_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => ? [X3: A] :
                ( member(A,X3,A4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A3)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa2),X3)
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_222_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => ? [X3: A] :
                ( member(A,X3,A4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X3)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa2)
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_223_finite_OemptyI,axiom,
    ! [A: $tType] : aa(set(A),$o,finite_finite2(A),bot_bot(set(A))) ).

% finite.emptyI
tff(fact_224_infinite__imp__nonempty,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
     => ( S != bot_bot(set(A)) ) ) ).

% infinite_imp_nonempty
tff(fact_225_rev__finite__subset,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
       => aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% rev_finite_subset
tff(fact_226_infinite__super,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
     => ( ~ aa(set(A),$o,finite_finite2(A),S)
       => ~ aa(set(A),$o,finite_finite2(A),T3) ) ) ).

% infinite_super
tff(fact_227_finite__subset,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% finite_subset
tff(fact_228_finite_OinsertI,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) ) ).

% finite.insertI
tff(fact_229_finite__UnI,axiom,
    ! [A: $tType,F3: set(A),G2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( aa(set(A),$o,finite_finite2(A),G2)
       => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),F3),G2)) ) ) ).

% finite_UnI
tff(fact_230_Un__infinite,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) ) ).

% Un_infinite
tff(fact_231_infinite__Un,axiom,
    ! [A: $tType,S: set(A),T3: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3))
    <=> ( ~ aa(set(A),$o,finite_finite2(A),S)
        | ~ aa(set(A),$o,finite_finite2(A),T3) ) ) ).

% infinite_Un
tff(fact_232_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,A4)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa2)
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_233_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,A4)
                & ! [Xa2: A] :
                    ( member(A,Xa2,A4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa2),X3)
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_234_finite_Ocases,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
     => ( ( A3 != bot_bot(set(A)) )
       => ~ ! [A7: set(A)] :
              ( ? [A5: A] : A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),A7)
             => ~ aa(set(A),$o,finite_finite2(A),A7) ) ) ) ).

% finite.cases
tff(fact_235_finite_Osimps,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A3)
    <=> ( ( A3 = bot_bot(set(A)) )
        | ? [A8: set(A),A9: A] :
            ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A9),A8) )
            & aa(set(A),$o,finite_finite2(A),A8) ) ) ) ).

% finite.simps
tff(fact_236_buildup__gives__valid,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Na),Na) ) ).

% buildup_gives_valid
tff(fact_237_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A3) ).

% bot_nat_0.extremum
tff(fact_238_le0,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% le0
tff(fact_239_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),Ya: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( member(A,Ya,S)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))),aa(A,B,F2,Ya)) ) ) ) ) ).

% arg_min_least
tff(fact_240_finite__transitivity__chain,axiom,
    ! [A: $tType,A4: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X3: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,X3),X3)
       => ( ! [X3: A,Y: A,Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),R,X3),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),R,Y),Z2)
               => aa(A,$o,aa(A,fun(A,$o),R,X3),Z2) ) )
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => ? [Y2: A] :
                    ( member(A,Y2,A4)
                    & aa(A,$o,aa(A,fun(A,$o),R,X3),Y2) ) )
           => ( A4 = bot_bot(set(A)) ) ) ) ) ) ).

% finite_transitivity_chain
tff(fact_241_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% deg_not_0
tff(fact_242_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_243_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_244_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => member(A,lattic7623131987881927897min_on(A,B,F2,S),S) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_245_Leaf__0__not,axiom,
    ! [A3: $o,B3: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A3),(B3)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_246_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),aa(set(A),A,lattic5882676163264333800up_fin(A),B2)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_247_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na)
        <=> ( Na = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_248_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),A3) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_249_neq0__conv,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% neq0_conv
tff(fact_250_less__nat__zero__code,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_251_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Sup_fin.singleton
tff(fact_252_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),B3: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y: A] :
            ( aa(A,$o,P,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y)),B3) )
       => ? [X3: A] :
            ( aa(A,$o,P,X3)
            & ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y2)),aa(A,nat,F2,X3)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_253_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,M: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,M,X3)),aa(A,nat,M,Y2)) ) ) ) ).

% ex_has_least_nat
tff(fact_254_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ).

% order_less_imp_not_less
tff(fact_255_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( Ya != Xa ) ) ) ).

% order_less_imp_not_eq2
tff(fact_256_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( Xa != Ya ) ) ) ).

% order_less_imp_not_eq
tff(fact_257_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
          | ( Xa = Ya )
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ).

% linorder_less_linear
tff(fact_258_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A,P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_259_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ).

% order_less_not_sym
tff(fact_260_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B3)),C3)
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C3) ) ) ) ) ).

% order_less_subst2
tff(fact_261_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,B3))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C3)) ) ) ) ) ).

% order_less_subst1
tff(fact_262_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Xa) ) ).

% order_less_irrefl
tff(fact_263_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( ( aa(A,B,F2,B3) = C3 )
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C3) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_264_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C3: B] :
          ( ( A3 = aa(B,A,F2,B3) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C3)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_265_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_less_trans
tff(fact_266_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% order_less_asym'
tff(fact_267_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa != Ya )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ) ).

% linorder_neq_iff
tff(fact_268_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ).

% order_less_asym
tff(fact_269_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa != Ya )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ) ).

% linorder_neqE
tff(fact_270_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( A3 != B3 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_271_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( A3 != B3 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_272_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A3) ) ) ) ).

% dual_order.strict_trans
tff(fact_273_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
            | ( Xa = Ya ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_274_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3) ) ) ) ).

% order.strict_trans
tff(fact_275_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B3: A] :
          ( ! [A5: A,B5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),B5)
             => aa(A,$o,aa(A,fun(A,$o),P,A5),B5) )
         => ( ! [A5: A] : aa(A,$o,aa(A,fun(A,$o),P,A5),A5)
           => ( ! [A5: A,B5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B5),A5)
                 => aa(A,$o,aa(A,fun(A,$o),P,A5),B5) )
             => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) ) ) ) ) ).

% linorder_less_wlog
tff(fact_276_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
        <=> ? [N: A] :
              ( aa(A,$o,P,N)
              & ! [M2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M2),N)
                 => ~ aa(A,$o,P,M2) ) ) ) ) ).

% exists_least_iff
tff(fact_277_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),A3) ) ).

% dual_order.irrefl
tff(fact_278_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% dual_order.asym
tff(fact_279_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( ( Xa != Ya )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ) ).

% linorder_cases
tff(fact_280_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ya: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
          <=> ( Xa = Ya ) ) ) ) ).

% antisym_conv3
tff(fact_281_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [X3: A] :
              ( ! [Y2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y2),X3)
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X3) )
         => aa(A,$o,P,A3) ) ) ).

% less_induct
tff(fact_282_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( ( B3 = C3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3) ) ) ) ).

% ord_less_eq_trans
tff(fact_283_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3) ) ) ) ).

% ord_eq_less_trans
tff(fact_284_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% order.asym
tff(fact_285_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( Xa != Ya ) ) ) ).

% less_imp_neq
tff(fact_286_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ? [Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),Ya) ) ) ) ).

% dense
tff(fact_287_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xa: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X_1) ) ).

% gt_ex
tff(fact_288_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] :
        ? [Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),Xa) ) ).

% lt_ex
tff(fact_289_nat__neq__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( M != Na )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ).

% nat_neq_iff
tff(fact_290_less__not__refl,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_not_refl
tff(fact_291_less__not__refl2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( M != Na ) ) ).

% less_not_refl2
tff(fact_292_less__not__refl3,axiom,
    ! [S3: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S3),Ta)
     => ( S3 != Ta ) ) ).

% less_not_refl3
tff(fact_293_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A3: A] :
          ( ! [X3: A] :
              ( ! [Y2: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X3))
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X3) )
         => aa(A,$o,P,A3) ) ) ).

% measure_induct
tff(fact_294_less__irrefl__nat,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),Na) ).

% less_irrefl_nat
tff(fact_295_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
             => aa(nat,$o,P,M3) )
         => aa(nat,$o,P,N2) )
     => aa(nat,$o,P,Na) ) ).

% nat_less_induct
tff(fact_296_infinite__descent,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N2: nat] :
          ( ~ aa(nat,$o,P,N2)
         => ? [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
              & ~ aa(nat,$o,P,M3) ) )
     => aa(nat,$o,P,Na) ) ).

% infinite_descent
tff(fact_297_linorder__neqE__nat,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( Xa != Ya )
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ya)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ya),Xa) ) ) ).

% linorder_neqE_nat
tff(fact_298_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A3: A] :
          ( ! [X3: A] :
              ( ! [Y2: A] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X3))
                 => aa(A,$o,P,Y2) )
             => aa(A,$o,P,X3) )
         => aa(A,$o,P,A3) ) ) ).

% measure_induct_rule
tff(fact_299_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V: fun(A,nat),Xa: A] :
      ( ! [X3: A] :
          ( ~ aa(A,$o,P,X3)
         => ? [Y2: A] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y2)),aa(A,nat,V,X3))
              & ~ aa(A,$o,P,Y2) ) )
     => aa(A,$o,P,Xa) ) ).

% infinite_descent_measure
tff(fact_300_le__refl,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),Na) ).

% le_refl
tff(fact_301_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K) ) ) ).

% le_trans
tff(fact_302_eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = Na )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% eq_imp_le
tff(fact_303_le__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( M = Na ) ) ) ).

% le_antisym
tff(fact_304_nat__less__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & ( M != Na ) ) ) ).

% nat_less_le
tff(fact_305_nat__le__linear,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ).

% nat_le_linear
tff(fact_306_less__imp__le__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_imp_le_nat
tff(fact_307_le__eq__less__or__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) ) ) ).

% le_eq_less_or_eq
tff(fact_308_less__or__eq__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_or_eq_imp_le
tff(fact_309_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y: nat] :
            ( aa(nat,$o,P,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),B3) )
       => ? [X3: nat] :
            ( aa(nat,$o,P,X3)
            & ! [Y2: nat] :
                ( aa(nat,$o,P,Y2)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y2),X3) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_310_le__neq__implies__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ( M != Na )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% le_neq_implies_less
tff(fact_311_less__mono__imp__le__mono,axiom,
    ! [F2: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,I2)),aa(nat,nat,F2,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F2,I)),aa(nat,nat,F2,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_312_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_313_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Na)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),K2)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_314_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_315_gr0I,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% gr0I
tff(fact_316_not__gr0,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ( Na = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_317_not__less0,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% not_less0
tff(fact_318_less__zeroE,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),zero_zero(nat)) ).

% less_zeroE
tff(fact_319_gr__implies__not0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( Na != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_320_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( ~ aa(nat,$o,P,N2)
             => ? [M3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N2)
                  & ~ aa(nat,$o,P,M3) ) ) )
       => aa(nat,$o,P,Na) ) ) ).

% infinite_descent0
tff(fact_321_infinite__descent0__measure,axiom,
    ! [A: $tType,V: fun(A,nat),P: fun(A,$o),Xa: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V,X3) = zero_zero(nat) )
         => aa(A,$o,P,X3) )
     => ( ! [X3: A] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,V,X3))
           => ( ~ aa(A,$o,P,X3)
             => ? [Y2: A] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,V,Y2)),aa(A,nat,V,X3))
                  & ~ aa(A,$o,P,Y2) ) ) )
       => aa(A,$o,P,Xa) ) ) ).

% infinite_descent0_measure
tff(fact_322_unbounded__k__infinite,axiom,
    ! [K: nat,S: set(nat)] :
      ( ! [M4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),M4)
         => ? [N3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N3)
              & member(nat,N3,S) ) )
     => ~ aa(set(nat),$o,finite_finite2(nat),S) ) ).

% unbounded_k_infinite
tff(fact_323_infinite__nat__iff__unbounded,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
    <=> ! [M2: nat] :
        ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
          & member(nat,N,S) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_324_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya) ) ) ).

% leD
tff(fact_325_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ).

% leI
tff(fact_326_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            | ( A3 = B3 ) ) ) ) ).

% nless_le
tff(fact_327_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
          <=> ( Xa = Ya ) ) ) ) ).

% antisym_conv1
tff(fact_328_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
          <=> ( Xa = Ya ) ) ) ) ).

% antisym_conv2
tff(fact_329_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Ya: A] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),X3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z) ) ) ).

% dense_ge
tff(fact_330_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Ya: A,Z: A] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Ya)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z) ) ) ).

% dense_le
tff(fact_331_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ) ).

% less_le_not_le
tff(fact_332_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ya: A,Xa: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya) ) ) ).

% not_le_imp_less
tff(fact_333_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
            | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
tff(fact_334_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
tff(fact_335_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3) ) ) ) ).

% order.strict_trans1
tff(fact_336_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3) ) ) ) ).

% order.strict_trans2
tff(fact_337_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% order.strict_iff_not
tff(fact_338_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Xa)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),W) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_339_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( ! [W: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),W)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W),Ya)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z) ) ) ) ).

% dense_le_bounded
tff(fact_340_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
            | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_341_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_342_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A3) ) ) ) ).

% dual_order.strict_trans1
tff(fact_343_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A3) ) ) ) ).

% dual_order.strict_trans2
tff(fact_344_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_345_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% order.strict_implies_order
tff(fact_346_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% dual_order.strict_implies_order
tff(fact_347_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
            | ( Xa = Ya ) ) ) ) ).

% order_le_less
tff(fact_348_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
            & ( Xa != Ya ) ) ) ) ).

% order_less_le
tff(fact_349_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ).

% linorder_not_le
tff(fact_350_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ).

% linorder_not_less
tff(fact_351_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ).

% order_less_imp_le
tff(fact_352_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( ( A3 != B3 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% order_le_neq_trans
tff(fact_353_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != B3 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% order_neq_le_trans
tff(fact_354_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_le_less_trans
tff(fact_355_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z) ) ) ) ).

% order_less_le_trans
tff(fact_356_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(B,A,F2,B3))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),B3),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X3)),aa(B,A,F2,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C3)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_357_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B3)),C3)
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C3) ) ) ) ) ).

% order_le_less_subst2
tff(fact_358_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,F2: fun(B,A),B3: B,C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,B3))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B3),C3)
           => ( ! [X3: B,Y: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,C3)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_359_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,F2: fun(A,B),C3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,B3)),C3)
           => ( ! [X3: A,Y: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),C3) ) ) ) ) ).

% order_less_le_subst2
tff(fact_360_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ).

% linorder_le_less_linear
tff(fact_361_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
            | ( Xa = Ya ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_362_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( ( Na != zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na) ) ) ).

% gr_zeroI
tff(fact_363_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Na),zero_zero(A)) ) ).

% not_less_zero
tff(fact_364_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na)
         => ( Na != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_365_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Na)
        <=> ( Na != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_366_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),A3) ) ) ).

% bot.not_eq_extremum
tff(fact_367_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_368_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.strict_coboundedI2
tff(fact_369_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% sup.strict_coboundedI1
tff(fact_370_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% sup.strict_order_iff
tff(fact_371_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B3),C3)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A3) ) ) ) ).

% sup.strict_boundedE
tff(fact_372_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = B3 ) ) ) ).

% sup.absorb4
tff(fact_373_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) = A3 ) ) ) ).

% sup.absorb3
tff(fact_374_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% less_supI2
tff(fact_375_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)) ) ) ).

% less_supI1
tff(fact_376_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ~ ? [X2: A] :
                  ( member(A,X2,S)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,F2,lattic7623131987881927897min_on(A,B,F2,S))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_377_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X4: set(A)] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => ? [Xa2: A] :
                    ( member(A,Xa2,X4)
                    & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xa2) ) )
           => ~ aa(set(A),$o,finite_finite2(A),X4) ) ) ) ).

% infinite_growing
tff(fact_378_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,S)
                & ~ ? [Xa2: A] :
                      ( member(A,Xa2,S)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa2),X3) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_379_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_380_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A4) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_381_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B5: A,A7: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A7)
                 => ( ! [X2: A] :
                        ( member(A,X2,A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),X2) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),A7)) ) ) )
             => aa(set(A),$o,P,A4) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_382_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B5: A,A7: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),A7)
                 => ( ! [X2: A] :
                        ( member(A,X2,A7)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),B5) )
                   => ( aa(set(A),$o,P,A7)
                     => aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),A7)) ) ) )
             => aa(set(A),$o,P,A4) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_383_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_384_le__0__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),zero_zero(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_385_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),zero_zero(nat))
     => ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_386_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),zero_zero(nat))
    <=> ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_387_less__eq__nat_Osimps_I1_J,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Na) ).

% less_eq_nat.simps(1)
tff(fact_388_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),Xa)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_389_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),Xa) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),Xa) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_390_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),Xa)
             => ! [A10: A] :
                  ( member(A,A10,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A10),Xa) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_391_infinite__nat__iff__unbounded__le,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
    <=> ! [M2: nat] :
        ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
          & member(nat,N,S) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_392_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),aa(set(A),A,lattic5882676163264333800up_fin(A),B2)) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_393_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( B2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B2)),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A4) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_394_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_395_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic5882676163264333800up_fin(A),A4),A4) ) ) ) ) ).

% Sup_fin.closed
tff(fact_396_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_397_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leaf((X21),(X22)) = vEBT_Leaf((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_398_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_399_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_400_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_401_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf((Uu),(Uv)),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_402_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A9: $o,B7: $o] : Ta = vEBT_Leaf((A9),(B7)) ) ).

% deg1Leaf
tff(fact_403_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
     => ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ).

% deg_1_Leaf
tff(fact_404_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( ( Na = one_one(nat) )
       => ? [A5: $o,B5: $o] : Ta = vEBT_Leaf((A5),(B5)) ) ) ).

% deg_1_Leafy
tff(fact_405_finite__nat__set__iff__bounded__le,axiom,
    ! [N4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N4)
    <=> ? [M2: nat] :
        ! [X: nat] :
          ( member(nat,X,N4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M2) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_406_bounded__nat__set__is__finite,axiom,
    ! [N4: set(nat),Na: nat] :
      ( ! [X3: nat] :
          ( member(nat,X3,N4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),Na) )
     => aa(set(nat),$o,finite_finite2(nat),N4) ) ).

% bounded_nat_set_is_finite
tff(fact_407_psubsetI,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( ( A4 != B2 )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2) ) ) ).

% psubsetI
tff(fact_408_less__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),one_one(nat))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_one
tff(fact_409_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xa: A] :
          ( ( one_one(A) = Xa )
        <=> ( Xa = one_one(A) ) ) ) ).

% one_reorient
tff(fact_410_not__psubset__empty,axiom,
    ! [A: $tType,A4: set(A)] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_411_finite__psubset__induct,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [A7: set(A)] :
            ( aa(set(A),$o,finite_finite2(A),A7)
           => ( ! [B8: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B8),A7)
                 => aa(set(A),$o,P,B8) )
             => aa(set(A),$o,P,A7) ) )
       => aa(set(A),$o,P,A4) ) ) ).

% finite_psubset_induct
tff(fact_412_psubsetE,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4) ) ) ).

% psubsetE
tff(fact_413_psubset__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
        & ( A4 != B2 ) ) ) ).

% psubset_eq
tff(fact_414_psubset__imp__subset,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ).

% psubset_imp_subset
tff(fact_415_psubset__subset__trans,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),C2) ) ) ).

% psubset_subset_trans
tff(fact_416_subset__not__subset__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4) ) ) ).

% subset_not_subset_eq
tff(fact_417_subset__psubset__trans,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B2),C2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),C2) ) ) ).

% subset_psubset_trans
tff(fact_418_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
        | ( A4 = B2 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_419_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_420_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_421_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less(fun(A,B)),F2),G)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F2),G)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F2) ) ) ) ).

% less_fun_def
tff(fact_422_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv)),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_423_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_424_vebt__member_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A3),(B3))),Xa)
    <=> $ite(
          Xa = zero_zero(nat),
          (A3),
          $ite(Xa = one_one(nat),(B3),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_425_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xa: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A3),(B3)),Xa)
    <=> $ite(
          Xa = zero_zero(nat),
          (A3),
          $ite(Xa = one_one(nat),(B3),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_426_bounded__Max__nat,axiom,
    ! [P: fun(nat,$o),Xa: nat,M5: nat] :
      ( aa(nat,$o,P,Xa)
     => ( ! [X3: nat] :
            ( aa(nat,$o,P,X3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M5) )
       => ~ ! [M4: nat] :
              ( aa(nat,$o,P,M4)
             => ~ ! [X2: nat] :
                    ( aa(nat,$o,P,X2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),M4) ) ) ) ) ).

% bounded_Max_nat
tff(fact_427_finite__nat__set__iff__bounded,axiom,
    ! [N4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),N4)
    <=> ? [M2: nat] :
        ! [X: nat] :
          ( member(nat,X,N4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),M2) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_428_vebt__insert_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xa: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A3),(B3)),Xa) = $ite(
        Xa = zero_zero(nat),
        vEBT_Leaf($true,(B3)),
        $ite(Xa = one_one(nat),vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) ).

% vebt_insert.simps(1)
tff(fact_429_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_430_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_431_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_432_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_433_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_434_nat__descend__induct,axiom,
    ! [Na: nat,P: fun(nat,$o),M: nat] :
      ( ! [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K2)
         => aa(nat,$o,P,K2) )
     => ( ! [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Na)
           => ( ! [I3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),I3)
                 => aa(nat,$o,P,I3) )
             => aa(nat,$o,P,K2) ) )
       => aa(nat,$o,P,M) ) ) ).

% nat_descend_induct
tff(fact_435_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_436_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D1)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D22)
           => ? [E: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D1)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),E),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_437_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,P,A3)
           => ( ~ aa(A,$o,P,B3)
             => ? [C5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C5)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C5),B3)
                  & ! [X2: A] :
                      ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X2)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),C5) )
                     => aa(A,$o,P,X2) )
                  & ! [D4: A] :
                      ( ! [X3: A] :
                          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X3)
                            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D4) )
                         => aa(A,$o,P,X3) )
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D4),C5) ) ) ) ) ) ) ).

% complete_interval
tff(fact_438_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B9: A,A11: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B9),A11)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A11),B9) ) ) ).

% verit_comp_simplify1(3)
tff(fact_439_psubset__trans,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),B2),C2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),C2) ) ) ).

% psubset_trans
tff(fact_440_psubsetD,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
     => ( member(A,C3,A4)
       => member(A,C3,B2) ) ) ).

% psubsetD
tff(fact_441_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),A3) ) ).

% verit_comp_simplify1(2)
tff(fact_442_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% verit_la_disequality
tff(fact_443_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),A3) ) ).

% verit_comp_simplify1(1)
tff(fact_444_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa != Ya )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_445_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A3: A] :
        ? [B5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B5)
          | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),A3) ) ) ).

% ex_gt_or_lt
tff(fact_446_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X2) ) ) ).

% minf(8)
tff(fact_447_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Ta) ) ) ).

% minf(6)
tff(fact_448_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X2) ) ) ).

% pinf(8)
tff(fact_449_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Ta) ) ) ).

% pinf(6)
tff(fact_450_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_451_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf((X21),(X22))) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_452_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_453_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A4) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Sup_fin.remove
tff(fact_454_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_455_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_456_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X22))) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_457_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A4: fun(B,A),B2: fun(B,A),Xa: B] : aa(B,A,aa(fun(B,A),fun(B,A),minus_minus(fun(B,A),A4),B2),Xa) = aa(A,A,minus_minus(A,aa(B,A,A4,Xa)),aa(B,A,B2,Xa)) ) ).

% minus_apply
tff(fact_458_DiffI,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,A4)
     => ( ~ member(A,C3,B2)
       => member(A,C3,aa(set(A),set(A),minus_minus(set(A),A4),B2)) ) ) ).

% DiffI
tff(fact_459_Diff__iff,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),minus_minus(set(A),A4),B2))
    <=> ( member(A,C3,A4)
        & ~ member(A,C3,B2) ) ) ).

% Diff_iff
tff(fact_460_Diff__idemp,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A4),B2)),B2) = aa(set(A),set(A),minus_minus(set(A),A4),B2) ).

% Diff_idemp
tff(fact_461_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),A3) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_462_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),zero_zero(A)) = A3 ) ).

% diff_zero
tff(fact_463_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_diff
tff(fact_464_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),zero_zero(A)) = A3 ) ).

% diff_0_right
tff(fact_465_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),A3) = zero_zero(A) ) ).

% diff_self
tff(fact_466_Diff__empty,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),bot_bot(set(A))) = A4 ).

% Diff_empty
tff(fact_467_empty__Diff,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),minus_minus(set(A),bot_bot(set(A))),A4) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_468_Diff__cancel,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),A4) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_469_finite__Diff2,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),A4),B2))
      <=> aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% finite_Diff2
tff(fact_470_finite__Diff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),A4),B2)) ) ).

% finite_Diff
tff(fact_471_Diff__insert0,axiom,
    ! [A: $tType,Xa: A,A4: set(A),B2: set(A)] :
      ( ~ member(A,Xa,A4)
     => ( aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),B2)) = aa(set(A),set(A),minus_minus(set(A),A4),B2) ) ) ).

% Diff_insert0
tff(fact_472_insert__Diff1,axiom,
    ! [A: $tType,Xa: A,B2: set(A),A4: set(A)] :
      ( member(A,Xa,B2)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),B2) = aa(set(A),set(A),minus_minus(set(A),A4),B2) ) ) ).

% insert_Diff1
tff(fact_473_Un__Diff__cancel,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),minus_minus(set(A),B2),A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) ).

% Un_Diff_cancel
tff(fact_474_Un__Diff__cancel2,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),B2),A4)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),A4) ).

% Un_Diff_cancel2
tff(fact_475_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% diff_ge_0_iff_ge
tff(fact_476_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% diff_gt_0_iff_gt
tff(fact_477_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_478_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A4),B2) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ).

% Diff_eq_empty_iff
tff(fact_479_insert__Diff__single,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4) ).

% insert_Diff_single
tff(fact_480_finite__Diff__insert,axiom,
    ! [A: $tType,A4: set(A),A3: A,B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)))
    <=> aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),A4),B2)) ) ).

% finite_Diff_insert
tff(fact_481_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Inf_fin.singleton
tff(fact_482_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),A3) = A3 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_483_DiffE,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),minus_minus(set(A),A4),B2))
     => ~ ( member(A,C3,A4)
         => member(A,C3,B2) ) ) ).

% DiffE
tff(fact_484_DiffD1,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),minus_minus(set(A),A4),B2))
     => member(A,C3,A4) ) ).

% DiffD1
tff(fact_485_DiffD2,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),minus_minus(set(A),A4),B2))
     => ~ member(A,C3,B2) ) ).

% DiffD2
tff(fact_486_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A4: fun(A,B),B2: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),minus_minus(fun(A,B),A4),B2),X2) = aa(B,B,minus_minus(B,aa(A,B,A4,X2)),aa(A,B,B2,X2)) ) ).

% fun_diff_def
tff(fact_487_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,minus_minus(A,C3),D2) )
         => ( ( A3 = B3 )
          <=> ( C3 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_488_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A3),C3)),B3) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A3),B3)),C3) ) ).

% diff_right_commute
tff(fact_489_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,nat,size_size(A),Xa) != aa(A,nat,size_size(A),Ya) )
         => ( Xa != Ya ) ) ) ).

% size_neq_size_imp_neq
tff(fact_490_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,minus_minus(A,C3),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_491_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A3),C3)),aa(A,A,minus_minus(A,B3),C3)) ) ) ).

% diff_right_mono
tff(fact_492_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,C3),A3)),aa(A,A,minus_minus(A,C3),B3)) ) ) ).

% diff_left_mono
tff(fact_493_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,D2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A3),C3)),aa(A,A,minus_minus(A,B3),D2)) ) ) ) ).

% diff_mono
tff(fact_494_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,A,minus_minus(A,A3),B3) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_495_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,D2: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),D2),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A3),C3)),aa(A,A,minus_minus(A,B3),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_496_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,minus_minus(A,C3),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_497_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,C3),A3)),aa(A,A,minus_minus(A,C3),B3)) ) ) ).

% diff_strict_left_mono
tff(fact_498_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A3),C3)),aa(A,A,minus_minus(A,B3),C3)) ) ) ).

% diff_strict_right_mono
tff(fact_499_Diff__infinite__finite,axiom,
    ! [A: $tType,T3: set(A),S: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),T3)
     => ( ~ aa(set(A),$o,finite_finite2(A),S)
       => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),S),T3)) ) ) ).

% Diff_infinite_finite
tff(fact_500_Diff__mono,axiom,
    ! [A: $tType,A4: set(A),C2: set(A),D3: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D3),B2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)),aa(set(A),set(A),minus_minus(set(A),C2),D3)) ) ) ).

% Diff_mono
tff(fact_501_Diff__subset,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)),A4) ).

% Diff_subset
tff(fact_502_double__diff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
       => ( aa(set(A),set(A),minus_minus(set(A),B2),aa(set(A),set(A),minus_minus(set(A),C2),A4)) = A4 ) ) ) ).

% double_diff
tff(fact_503_insert__Diff__if,axiom,
    ! [A: $tType,Xa: A,A4: set(A),B2: set(A)] :
      aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),B2) = $ite(member(A,Xa,B2),aa(set(A),set(A),minus_minus(set(A),A4),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(set(A),set(A),minus_minus(set(A),A4),B2))) ).

% insert_Diff_if
tff(fact_504_Un__Diff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)),C2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),C2)),aa(set(A),set(A),minus_minus(set(A),B2),C2)) ).

% Un_Diff
tff(fact_505_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
     => ? [B5: A] : member(A,B5,aa(set(A),set(A),minus_minus(set(A),B2),A4)) ) ).

% psubset_imp_ex_mem
tff(fact_506_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A3),B3)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_507_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A3),B3)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_508_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,minus_minus(A,Xa),Ya) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ).

% diff_shunt_var
tff(fact_509_Diff__insert,axiom,
    ! [A: $tType,A4: set(A),A3: A,B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A4),B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_510_insert__Diff,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( member(A,A3,A4)
     => ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = A4 ) ) ).

% insert_Diff
tff(fact_511_Diff__insert2,axiom,
    ! [A: $tType,A4: set(A),A3: A,B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),B2) ).

% Diff_insert2
tff(fact_512_Diff__insert__absorb,axiom,
    ! [A: $tType,Xa: A,A4: set(A)] :
      ( ~ member(A,Xa,A4)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = A4 ) ) ).

% Diff_insert_absorb
tff(fact_513_subset__Diff__insert,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),Xa: A,C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),minus_minus(set(A),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),C2)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),minus_minus(set(A),B2),C2))
        & ~ member(A,Xa,A4) ) ) ).

% subset_Diff_insert
tff(fact_514_Diff__subset__conv,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)),C2)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2)) ) ).

% Diff_subset_conv
tff(fact_515_Diff__partition,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),minus_minus(set(A),B2),A4)) = B2 ) ) ).

% Diff_partition
tff(fact_516_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),A3) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_517_infinite__remove,axiom,
    ! [A: $tType,S: set(A),A3: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_518_infinite__coinduct,axiom,
    ! [A: $tType,X4: fun(set(A),$o),A4: set(A)] :
      ( aa(set(A),$o,X4,A4)
     => ( ! [A7: set(A)] :
            ( aa(set(A),$o,X4,A7)
           => ? [X2: A] :
                ( member(A,X2,A7)
                & ( aa(set(A),$o,X4,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A)))))
                  | ~ aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))) ) ) )
       => ~ aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% infinite_coinduct
tff(fact_519_finite__empty__induct,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,P,A4)
       => ( ! [A5: A,A7: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A7)
             => ( member(A,A5,A7)
               => ( aa(set(A),$o,P,A7)
                 => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A5),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_520_subset__insert__iff,axiom,
    ! [A: $tType,A4: set(A),Xa: A,B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),B2))
    <=> $ite(member(A,Xa,A4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),B2),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)) ) ).

% subset_insert_iff
tff(fact_521_Diff__single__insert,axiom,
    ! [A: $tType,A4: set(A),Xa: A,B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),B2)) ) ).

% Diff_single_insert
tff(fact_522_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
             => ( aa(A,$o,P,X3)
              <=> aa(A,$o,P2,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q2,X3) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
               => ( ( aa(A,$o,P,X2)
                    & aa(A,$o,Q,X2) )
                <=> ( aa(A,$o,P2,X2)
                    & aa(A,$o,Q2,X2) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_523_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
             => ( aa(A,$o,P,X3)
              <=> aa(A,$o,P2,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z3),X3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q2,X3) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
               => ( ( aa(A,$o,P,X2)
                    | aa(A,$o,Q,X2) )
                <=> ( aa(A,$o,P2,X2)
                    | aa(A,$o,Q2,X2) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_524_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => ( X2 != Ta ) ) ) ).

% pinf(3)
tff(fact_525_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => ( X2 != Ta ) ) ) ).

% pinf(4)
tff(fact_526_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Ta) ) ) ).

% pinf(5)
tff(fact_527_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X2) ) ) ).

% pinf(7)
tff(fact_528_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => ( F3 = F3 ) ) ) ).

% pinf(11)
tff(fact_529_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
             => ( aa(A,$o,P,X3)
              <=> aa(A,$o,P2,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q2,X3) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
               => ( ( aa(A,$o,P,X2)
                    & aa(A,$o,Q,X2) )
                <=> ( aa(A,$o,P2,X2)
                    & aa(A,$o,Q2,X2) ) ) ) ) ) ) ).

% minf(1)
tff(fact_530_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P2: fun(A,$o),Q: fun(A,$o),Q2: fun(A,$o)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
             => ( aa(A,$o,P,X3)
              <=> aa(A,$o,P2,X3) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z3)
               => ( aa(A,$o,Q,X3)
                <=> aa(A,$o,Q2,X3) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
               => ( ( aa(A,$o,P,X2)
                    | aa(A,$o,Q,X2) )
                <=> ( aa(A,$o,P2,X2)
                    | aa(A,$o,Q2,X2) ) ) ) ) ) ) ).

% minf(2)
tff(fact_531_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => ( X2 != Ta ) ) ) ).

% minf(3)
tff(fact_532_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => ( X2 != Ta ) ) ) ).

% minf(4)
tff(fact_533_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Ta) ) ) ).

% minf(5)
tff(fact_534_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),X2) ) ) ).

% minf(7)
tff(fact_535_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F3: B] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => ( F3 = F3 ) ) ) ).

% minf(11)
tff(fact_536_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4))
             => ! [A10: A] :
                  ( member(A,A10,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A10) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_537_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A5) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_538_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4))
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_539_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B2: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ aa(set(A),$o,finite_finite2(A),B2)
         => aa(set(A),$o,P,B2) )
       => ( ! [A7: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B2)
                 => ( ! [X2: A] :
                        ( member(A,X2,A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B2) ) ) ) ).

% remove_induct
tff(fact_540_finite__remove__induct,axiom,
    ! [A: $tType,B2: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [A7: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),A7)
             => ( ( A7 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),B2)
                 => ( ! [X2: A] :
                        ( member(A,X2,A7)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A7) ) ) ) )
         => aa(set(A),$o,P,B2) ) ) ) ).

% finite_remove_induct
tff(fact_541_finite__induct__select,axiom,
    ! [A: $tType,S: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T4: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),T4),S)
             => ( aa(set(A),$o,P,T4)
               => ? [X2: A] :
                    ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),S),T4))
                    & aa(set(A),$o,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X2),T4)) ) ) )
         => aa(set(A),$o,P,S) ) ) ) ).

% finite_induct_select
tff(fact_542_psubset__insert__iff,axiom,
    ! [A: $tType,A4: set(A),Xa: A,B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),B2))
    <=> $ite(
          member(A,Xa,B2),
          aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2),
          $ite(member(A,Xa,A4),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),B2),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)) ) ) ).

% psubset_insert_iff
tff(fact_543_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B2)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_544_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_545_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arsinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_546_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_547_remove__def,axiom,
    ! [A: $tType,Xa: A,A4: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Xa),A4) = aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ).

% remove_def
tff(fact_548_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_549_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Inf_fin.remove
tff(fact_550_card__Diff1__less__iff,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4))
    <=> ( aa(set(A),$o,finite_finite2(A),A4)
        & member(A,Xa,A4) ) ) ).

% card_Diff1_less_iff
tff(fact_551_card__Diff2__less,axiom,
    ! [A: $tType,A4: set(A),Xa: A,Ya: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( member(A,Xa,A4)
       => ( member(A,Ya,A4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)) ) ) ) ).

% card_Diff2_less
tff(fact_552_card__Diff1__less,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( member(A,Xa,A4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)) ) ) ).

% card_Diff1_less
tff(fact_553_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_554_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% of_nat_0_less_iff
tff(fact_555_finite__Int,axiom,
    ! [A: $tType,F3: set(A),G2: set(A)] :
      ( ( aa(set(A),$o,finite_finite2(A),F3)
        | aa(set(A),$o,finite_finite2(A),G2) )
     => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),F3),G2)) ) ).

% finite_Int
tff(fact_556_diff__0__eq__0,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,zero_zero(nat)),Na) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_557_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_558_Int__subset__iff,axiom,
    ! [A: $tType,C2: set(A),A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),A4)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),B2) ) ) ).

% Int_subset_iff
tff(fact_559_inf__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(B,A),G: fun(B,A),Xa: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),inf_inf(fun(B,A)),F2),G),Xa) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,Xa)),aa(B,A,G,Xa)) ) ).

% inf_apply
tff(fact_560_inf__right__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),Ya) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) ) ).

% inf_right_idem
tff(fact_561_inf_Oright__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ).

% inf.right_idem
tff(fact_562_inf__left__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) ) ).

% inf_left_idem
tff(fact_563_inf_Oleft__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ).

% inf.left_idem
tff(fact_564_inf__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Xa) = Xa ) ).

% inf_idem
tff(fact_565_inf_Oidem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),A3) = A3 ) ).

% inf.idem
tff(fact_566_Int__insert__right__if1,axiom,
    ! [A: $tType,A3: A,A4: set(A),B2: set(A)] :
      ( member(A,A3,A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_567_Int__insert__right__if0,axiom,
    ! [A: $tType,A3: A,A4: set(A),B2: set(A)] :
      ( ~ member(A,A3,A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) ) ) ).

% Int_insert_right_if0
tff(fact_568_insert__inter__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A),B2: 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(A,fun(set(A),set(A)),insert2(A),A3),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) ).

% insert_inter_insert
tff(fact_569_Int__insert__left__if1,axiom,
    ! [A: $tType,A3: A,C2: set(A),B2: set(A)] :
      ( member(A,A3,C2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)),C2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_570_Int__insert__left__if0,axiom,
    ! [A: $tType,A3: A,C2: set(A),B2: set(A)] :
      ( ~ member(A,A3,C2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)),C2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_571_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( M = Na ) ) ) ).

% of_nat_eq_iff
tff(fact_572_diff__diff__cancel,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),Na)
     => ( aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,minus_minus(nat,Na),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_573_Un__Int__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T3: 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),T3)),S) = S ).

% Un_Int_eq(1)
tff(fact_574_Un__Int__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T3: 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),T3)),T3) = T3 ).

% Un_Int_eq(2)
tff(fact_575_Un__Int__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T3: 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),T3)) = S ).

% Un_Int_eq(3)
tff(fact_576_Un__Int__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = T3 ).

% Un_Int_eq(4)
tff(fact_577_Int__Un__eq_I1_J,axiom,
    ! [A: $tType,S: set(A),T3: 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),T3)),S) = S ).

% Int_Un_eq(1)
tff(fact_578_Int__Un__eq_I2_J,axiom,
    ! [A: $tType,S: set(A),T3: 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),T3)),T3) = T3 ).

% Int_Un_eq(2)
tff(fact_579_Int__Un__eq_I3_J,axiom,
    ! [A: $tType,S: set(A),T3: 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),T3)) = S ).

% Int_Un_eq(3)
tff(fact_580_Int__Un__eq_I4_J,axiom,
    ! [A: $tType,T3: set(A),S: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),T3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3)) = T3 ).

% Int_Un_eq(4)
tff(fact_581_member__remove,axiom,
    ! [A: $tType,Xa: A,Ya: A,A4: set(A)] :
      ( member(A,Xa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),Ya),A4))
    <=> ( member(A,Xa,A4)
        & ( Xa != Ya ) ) ) ).

% member_remove
tff(fact_582_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3) ) ) ) ).

% inf.bounded_iff
tff(fact_583_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z) ) ) ) ).

% le_inf_iff
tff(fact_584_zero__less__diff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,minus_minus(nat,Na),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% zero_less_diff
tff(fact_585_inf__bot__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% inf_bot_right
tff(fact_586_inf__bot__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% inf_bot_left
tff(fact_587_boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_right
tff(fact_588_boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% boolean_algebra.conj_zero_left
tff(fact_589_insert__disjoint_I1_J,axiom,
    ! [A: $tType,A3: A,A4: set(A),B2: 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(A,fun(set(A),set(A)),insert2(A),A3),A4)),B2) = bot_bot(set(A)) )
    <=> ( ~ member(A,A3,B2)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) ) ) ) ).

% insert_disjoint(1)
tff(fact_590_insert__disjoint_I2_J,axiom,
    ! [A: $tType,A3: A,A4: set(A),B2: 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),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)),B2) )
    <=> ( ~ member(A,A3,B2)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_591_disjoint__insert_I1_J,axiom,
    ! [A: $tType,B2: set(A),A3: A,A4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = bot_bot(set(A)) )
    <=> ( ~ member(A,A3,B2)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),A4) = bot_bot(set(A)) ) ) ) ).

% disjoint_insert(1)
tff(fact_592_disjoint__insert_I2_J,axiom,
    ! [A: $tType,A4: set(A),B3: A,B2: set(A)] :
      ( ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),B2)) )
    <=> ( ~ member(A,B3,A4)
        & ( bot_bot(set(A)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_593_diff__is__0__eq_H,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_594_diff__is__0__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% diff_is_0_eq
tff(fact_595_sup__inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)) = Xa ) ).

% sup_inf_absorb
tff(fact_596_inf__sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) = Xa ) ).

% inf_sup_absorb
tff(fact_597_Diff__disjoint,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),minus_minus(set(A),B2),A4)) = bot_bot(set(A)) ).

% Diff_disjoint
tff(fact_598_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_599_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( zero_zero(nat) = Na ) ) ) ).

% of_nat_0_eq_iff
tff(fact_600_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_601_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% of_nat_less_iff
tff(fact_602_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% of_nat_le_iff
tff(fact_603_card_Oempty,axiom,
    ! [A: $tType] : aa(set(A),nat,finite_card(A),bot_bot(set(A))) = zero_zero(nat) ).

% card.empty
tff(fact_604_card_Oinfinite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),nat,finite_card(A),A4) = zero_zero(nat) ) ) ).

% card.infinite
tff(fact_605_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Na) = one_one(A) )
        <=> ( Na = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_606_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),Na) )
        <=> ( Na = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_607_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_608_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) = A3 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_609_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),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_610_card__0__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( aa(set(A),nat,finite_card(A),A4) = zero_zero(nat) )
      <=> ( A4 = bot_bot(set(A)) ) ) ) ).

% card_0_eq
tff(fact_611_card__Diff__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A),B2: set(A)] :
      ( member(A,A3,A4)
     => ( ~ member(A,A3,B2)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2))) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),B2))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_612_boolean__algebra__cancel_Oinf2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,K: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),B3) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B2) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)) ) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_613_boolean__algebra__cancel_Oinf1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: A,K: A,A3: A,B3: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),K),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)) ) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_614_inf__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( semilattice_inf(B)
     => ! [F2: fun(A,B),G: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),inf_inf(fun(A,B)),F2),G),X2) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) ) ).

% inf_fun_def
tff(fact_615_inf__left__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ).

% inf_left_commute
tff(fact_616_inf_Oleft__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C3)) ) ).

% inf.left_commute
tff(fact_617_inf__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Xa) ) ).

% inf_commute
tff(fact_618_inf_Ocommute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),A3) ) ).

% inf.commute
tff(fact_619_inf__assoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)) ) ).

% inf_assoc
tff(fact_620_inf_Oassoc,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C3)) ) ).

% inf.assoc
tff(fact_621_inf__sup__aci_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Xa) ) ).

% inf_sup_aci(1)
tff(fact_622_inf__sup__aci_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),Z) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)) ) ).

% inf_sup_aci(2)
tff(fact_623_inf__sup__aci_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ).

% inf_sup_aci(3)
tff(fact_624_inf__sup__aci_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) ) ).

% inf_sup_aci(4)
tff(fact_625_card__Diff__subset__Int,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ).

% card_Diff_subset_Int
tff(fact_626_zdiff__int__split,axiom,
    ! [P: fun(int,$o),Xa: nat,Ya: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,Xa),Ya)))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ya),Xa)
         => aa(int,$o,P,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Xa)),aa(nat,int,semiring_1_of_nat(int),Ya))) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ya)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_627_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),C3) ) ) ).

% inf.coboundedI2
tff(fact_628_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),C3) ) ) ).

% inf.coboundedI1
tff(fact_629_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = B3 ) ) ) ).

% inf.absorb_iff2
tff(fact_630_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = A3 ) ) ) ).

% inf.absorb_iff1
tff(fact_631_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),B3) ) ).

% inf.cobounded2
tff(fact_632_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),A3) ) ).

% inf.cobounded1
tff(fact_633_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ) ) ).

% inf.order_iff
tff(fact_634_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)) ) ) ) ).

% inf_greatest
tff(fact_635_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C3)) ) ) ) ).

% inf.boundedI
tff(fact_636_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3) ) ) ) ).

% inf.boundedE
tff(fact_637_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) = Ya ) ) ) ).

% inf_absorb2
tff(fact_638_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) = Xa ) ) ) ).

% inf_absorb1
tff(fact_639_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = B3 ) ) ) ).

% inf.absorb2
tff(fact_640_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = A3 ) ) ) ).

% inf.absorb1
tff(fact_641_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) = Xa ) ) ) ).

% le_iff_inf
tff(fact_642_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F2: fun(A,fun(A,A)),Xa: A,Ya: A] :
          ( ! [X3: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),Y)),X3)
         => ( ! [X3: A,Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),Y)),Y)
           => ( ! [X3: A,Y: A,Z2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2)
                   => aa(A,$o,aa(A,fun(A,$o),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),Xa),Ya) = aa(A,A,aa(A,fun(A,A),F2,Xa),Ya) ) ) ) ) ) ).

% inf_unique
tff(fact_643_inf_OorderI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% inf.orderI
tff(fact_644_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ) ) ).

% inf.orderE
tff(fact_645_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,Xa: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),Xa) ) ) ).

% le_infI2
tff(fact_646_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,Xa: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),Xa) ) ) ).

% le_infI1
tff(fact_647_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C3: A,B3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C3),D2)) ) ) ) ).

% inf_mono
tff(fact_648_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)) ) ) ) ).

% le_infI
tff(fact_649_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),B3) ) ) ) ).

% le_infE
tff(fact_650_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),Ya) ) ).

% inf_le2
tff(fact_651_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),Xa) ) ).

% inf_le1
tff(fact_652_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),Xa) ) ).

% inf_sup_ord(1)
tff(fact_653_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),Ya) ) ).

% inf_sup_ord(2)
tff(fact_654_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),C3) ) ) ).

% inf.strict_coboundedI2
tff(fact_655_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),C3) ) ) ).

% inf.strict_coboundedI1
tff(fact_656_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% inf.strict_order_iff
tff(fact_657_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),inf_inf(A),B3),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3) ) ) ) ).

% inf.strict_boundedE
tff(fact_658_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = B3 ) ) ) ).

% inf.absorb4
tff(fact_659_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = A3 ) ) ) ).

% inf.absorb3
tff(fact_660_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B3: A,Xa: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),Xa) ) ) ).

% less_infI2
tff(fact_661_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: A,Xa: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3)),Xa) ) ) ).

% less_infI1
tff(fact_662_sup__inf__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Ya: A,Z: A,Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)),Xa) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Xa)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z),Xa)) ) ).

% sup_inf_distrib2
tff(fact_663_sup__inf__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ).

% sup_inf_distrib1
tff(fact_664_inf__sup__distrib2,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Ya: A,Z: A,Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)),Xa) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z),Xa)) ) ).

% inf_sup_distrib2
tff(fact_665_inf__sup__distrib1,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ).

% inf_sup_distrib1
tff(fact_666_distrib__imp2,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( ! [X3: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Z2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ) ) ).

% distrib_imp2
tff(fact_667_distrib__imp1,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( ! [X3: A,Y: A,Z2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Z2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ) ) ).

% distrib_imp1
tff(fact_668_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Ya: A,Z: A,Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)),Xa) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Xa)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Z),Xa)) ) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_669_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Ya: A,Z: A,Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)),Xa) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Z),Xa)) ) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_670_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z)) ) ).

% boolean_algebra.disj_conj_distrib
tff(fact_671_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z)) ) ).

% boolean_algebra.conj_disj_distrib
tff(fact_672_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_673_nat__int__comparison_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ) ).

% nat_int_comparison(2)
tff(fact_674_disjoint__iff__not__equal,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
    <=> ! [X: A] :
          ( member(A,X,A4)
         => ! [Xa3: A] :
              ( member(A,Xa3,B2)
             => ( X != Xa3 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_675_Int__empty__right,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),bot_bot(set(A))) = bot_bot(set(A)) ).

% Int_empty_right
tff(fact_676_Int__empty__left,axiom,
    ! [A: $tType,B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),bot_bot(set(A))),B2) = bot_bot(set(A)) ).

% Int_empty_left
tff(fact_677_disjoint__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
    <=> ! [X: A] :
          ( member(A,X,A4)
         => ~ member(A,X,B2) ) ) ).

% disjoint_iff
tff(fact_678_Int__emptyI,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ! [X3: A] :
          ( member(A,X3,A4)
         => ~ member(A,X3,B2) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) ) ) ).

% Int_emptyI
tff(fact_679_Int__Collect__mono,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( ! [X3: A] :
            ( member(A,X3,A4)
           => ( aa(A,$o,P,X3)
             => aa(A,$o,Q,X3) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),aa(fun(A,$o),set(A),collect(A),Q))) ) ) ).

% Int_Collect_mono
tff(fact_680_Int__greatest,axiom,
    ! [A: $tType,C2: set(A),A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),B2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) ) ) ).

% Int_greatest
tff(fact_681_Int__absorb2,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = A4 ) ) ).

% Int_absorb2
tff(fact_682_Int__absorb1,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_683_Int__lower2,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),B2) ).

% Int_lower2
tff(fact_684_Int__lower1,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),A4) ).

% Int_lower1
tff(fact_685_Int__mono,axiom,
    ! [A: $tType,A4: set(A),C2: set(A),B2: set(A),D3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),D3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C2),D3)) ) ) ).

% Int_mono
tff(fact_686_nat__int__comparison_I3_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ) ).

% nat_int_comparison(3)
tff(fact_687_Int__insert__right,axiom,
    ! [A: $tType,A4: set(A),A3: A,B2: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2)) = $ite(member(A,A3,A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) ).

% Int_insert_right
tff(fact_688_Int__insert__left,axiom,
    ! [A: $tType,A3: A,B2: set(A),C2: 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(A,fun(set(A),set(A)),insert2(A),A3),B2)),C2) = $ite(member(A,A3,C2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)) ).

% Int_insert_left
tff(fact_689_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_690_diffs0__imp__equal,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Na) = zero_zero(nat) )
     => ( ( aa(nat,nat,minus_minus(nat,Na),M) = zero_zero(nat) )
       => ( M = Na ) ) ) ).

% diffs0_imp_equal
tff(fact_691_diff__less__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,L),Na)),aa(nat,nat,minus_minus(nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_692_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,J),Na)),K) ) ).

% less_imp_diff_less
tff(fact_693_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,Na),K) )
        <=> ( M = Na ) ) ) ) ).

% eq_diff_iff
tff(fact_694_le__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% le_diff_iff
tff(fact_695_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K)) = aa(nat,nat,minus_minus(nat,M),Na) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_696_diff__le__mono,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,Na),L)) ) ).

% diff_le_mono
tff(fact_697_diff__le__self,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Na)),M) ).

% diff_le_self
tff(fact_698_le__diff__iff_H,axiom,
    ! [A3: nat,C3: nat,B3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),C3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B3),C3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,C3),A3)),aa(nat,nat,minus_minus(nat,C3),B3))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B3),A3) ) ) ) ).

% le_diff_iff'
tff(fact_699_diff__le__mono2,axiom,
    ! [M: nat,Na: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,L),Na)),aa(nat,nat,minus_minus(nat,L),M)) ) ).

% diff_le_mono2
tff(fact_700_Un__Int__distrib2,axiom,
    ! [A: $tType,B2: set(A),C2: set(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)),A4) = 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)),B2),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C2),A4)) ).

% Un_Int_distrib2
tff(fact_701_Int__Un__distrib2,axiom,
    ! [A: $tType,B2: set(A),C2: set(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2)),A4) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C2),A4)) ).

% Int_Un_distrib2
tff(fact_702_Un__Int__distrib,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)) = 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)),A4),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C2)) ).

% Un_Int_distrib
tff(fact_703_Int__Un__distrib,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C2)) ).

% Int_Un_distrib
tff(fact_704_Un__Int__crazy,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: 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)),A4),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C2),A4)) = 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)),A4),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C2),A4)) ).

% Un_Int_crazy
tff(fact_705_Diff__Int__distrib2,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)),C2) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)) ).

% Diff_Int_distrib2
tff(fact_706_Diff__Int__distrib,axiom,
    ! [A: $tType,C2: set(A),A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C2),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = aa(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)),C2),A4)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C2),B2)) ).

% Diff_Int_distrib
tff(fact_707_Diff__Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) ).

% Diff_Diff_Int
tff(fact_708_Diff__Int2,axiom,
    ! [A: $tType,A4: set(A),C2: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C2)),B2) ).

% Diff_Int2
tff(fact_709_Int__Diff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),C2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),minus_minus(set(A),B2),C2)) ).

% Int_Diff
tff(fact_710_card__Diff__subset,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2)) ) ) ) ).

% card_Diff_subset
tff(fact_711_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),B2))) ) ).

% diff_card_le_card_Diff
tff(fact_712_card__Diff__singleton__if,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))) = $ite(member(A,Xa,A4),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A4)),one_one(nat)),aa(set(A),nat,finite_card(A),A4)) ).

% card_Diff_singleton_if
tff(fact_713_card__Diff__singleton,axiom,
    ! [A: $tType,Xa: A,A4: set(A)] :
      ( member(A,Xa,A4)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A4)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_714_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% of_nat_0_le_iff
tff(fact_715_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_716_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z))) ) ).

% distrib_inf_le
tff(fact_717_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Z))) ) ).

% distrib_sup_le
tff(fact_718_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ) ).

% less_imp_of_nat_less
tff(fact_719_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% of_nat_less_imp_less
tff(fact_720_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_721_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ? [B4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B4)
          & ( aa(set(A),nat,finite_card(A),B4) = Na )
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),A4) ) ) ).

% infinite_arbitrarily_large
tff(fact_722_card__subset__eq,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
       => ( ( aa(set(A),nat,finite_card(A),A4) = aa(set(A),nat,finite_card(A),B2) )
         => ( A4 = B2 ) ) ) ) ).

% card_subset_eq
tff(fact_723_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B2: set(A),A4: set(B),R2: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( ! [A5: B] :
            ( member(B,A5,A4)
           => ? [B10: A] :
                ( member(A,B10,B2)
                & aa(A,$o,aa(B,fun(A,$o),R2,A5),B10) ) )
       => ( ! [A1: B,A22: B,B5: A] :
              ( member(B,A1,A4)
             => ( member(B,A22,A4)
               => ( member(A,B5,B2)
                 => ( aa(A,$o,aa(B,fun(A,$o),R2,A1),B5)
                   => ( aa(A,$o,aa(B,fun(A,$o),R2,A22),B5)
                     => ( A1 = A22 ) ) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A4)),aa(set(A),nat,finite_card(A),B2)) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_724_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,M),Na)) = aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Na)) ) ) ) ).

% of_nat_diff
tff(fact_725_card__insert__le,axiom,
    ! [A: $tType,A4: set(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4))) ).

% card_insert_le
tff(fact_726_Diff__triv,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
     => ( aa(set(A),set(A),minus_minus(set(A),A4),B2) = A4 ) ) ).

% Diff_triv
tff(fact_727_Int__Diff__disjoint,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = bot_bot(set(A)) ).

% Int_Diff_disjoint
tff(fact_728_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),C2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),A4) ) ).

% Un_Int_assoc_eq
tff(fact_729_diff__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),Na)),M) ) ) ).

% diff_less
tff(fact_730_less__diff__iff,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Na),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% less_diff_iff
tff(fact_731_diff__less__mono,axiom,
    ! [A3: nat,B3: nat,C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C3),A3)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,A3),C3)),aa(nat,nat,minus_minus(nat,B3),C3)) ) ) ).

% diff_less_mono
tff(fact_732_Diff__Un,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),C2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)),aa(set(A),set(A),minus_minus(set(A),A4),C2)) ).

% Diff_Un
tff(fact_733_Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)),aa(set(A),set(A),minus_minus(set(A),A4),C2)) ).

% Diff_Int
tff(fact_734_Int__Diff__Un,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = A4 ).

% Int_Diff_Un
tff(fact_735_Un__Diff__Int,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) = A4 ).

% Un_Diff_Int
tff(fact_736_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_737_card__insert__le__m1,axiom,
    ! [A: $tType,Na: nat,Ya: set(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Ya)),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),Ya))),Na) ) ) ).

% card_insert_le_m1
tff(fact_738_is__singleton__altdef,axiom,
    ! [A: $tType,A4: set(A)] :
      ( is_singleton(A,A4)
    <=> ( aa(set(A),nat,finite_card(A),A4) = one_one(nat) ) ) ).

% is_singleton_altdef
tff(fact_739_card__eq__0__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = zero_zero(nat) )
    <=> ( ( A4 = bot_bot(set(A)) )
        | ~ aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% card_eq_0_iff
tff(fact_740_card__ge__0__finite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4))
     => aa(set(A),$o,finite_finite2(A),A4) ) ).

% card_ge_0_finite
tff(fact_741_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F3: set(A),C2: nat] :
      ( ! [G3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),G3),F3)
         => ( aa(set(A),$o,finite_finite2(A),G3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G3)),C2) ) )
     => ( aa(set(A),$o,finite_finite2(A),F3)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F3)),C2) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_742_card__seteq,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B2)),aa(set(A),nat,finite_card(A),A4))
         => ( A4 = B2 ) ) ) ) ).

% card_seteq
tff(fact_743_card__mono,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2)) ) ) ).

% card_mono
tff(fact_744_obtain__subset__with__card__n,axiom,
    ! [A: $tType,Na: nat,S: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(set(A),nat,finite_card(A),S))
     => ~ ! [T4: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T4),S)
           => ( ( aa(set(A),nat,finite_card(A),T4) = Na )
             => ~ aa(set(A),$o,finite_finite2(A),T4) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_745_card__less__sym__Diff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),B2))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),B2),A4))) ) ) ) ).

% card_less_sym_Diff
tff(fact_746_card__le__sym__Diff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),B2))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),B2),A4))) ) ) ) ).

% card_le_sym_Diff
tff(fact_747_card__1__singletonE,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = one_one(nat) )
     => ~ ! [X3: A] : A4 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_748_psubset__card__mono,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2)) ) ) ).

% psubset_card_mono
tff(fact_749_card__gt__0__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4))
    <=> ( ( A4 != bot_bot(set(A)) )
        & aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% card_gt_0_iff
tff(fact_750_card__Diff1__le,axiom,
    ! [A: $tType,A4: set(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A4)) ).

% card_Diff1_le
tff(fact_751_card__psubset,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2))
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2) ) ) ) ).

% card_psubset
tff(fact_752_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( B2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B2)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_753_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_754_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic7752659483105999362nf_fin(A),A4),A4) ) ) ) ) ).

% Inf_fin.closed
tff(fact_755_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B2)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_756_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ~ ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ).

% pos_int_cases
tff(fact_757_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ? [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_758_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% reals_Archimedean2
tff(fact_759_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% real_arch_simple
tff(fact_760_finite__enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),M: nat,Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(set(A),nat,finite_card(A),S))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S))
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S),M)),aa(nat,A,infini527867602293511546merate(A,S),Na))
              <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ) ) ).

% finite_enumerate_mono_iff
tff(fact_761_finite__enum__subset,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X4: set(A),Y3: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X4))
             => ( aa(nat,A,infini527867602293511546merate(A,X4),I2) = aa(nat,A,infini527867602293511546merate(A,Y3),I2) ) )
         => ( aa(set(A),$o,finite_finite2(A),X4)
           => ( aa(set(A),$o,finite_finite2(A),Y3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),X4)),aa(set(A),nat,finite_card(A),Y3))
               => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),Y3) ) ) ) ) ) ).

% finite_enum_subset
tff(fact_762_card_Oremove,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( member(A,Xa,A4)
       => ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_763_card_Oinsert__remove,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_764_card__Suc__Diff1,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( member(A,Xa,A4)
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A4) ) ) ) ).

% card_Suc_Diff1
tff(fact_765_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => ( ( Na != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),M))),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Na))) ) ) ) ).

% inverse_of_nat_le
tff(fact_766_nat_Oinject,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( aa(nat,nat,suc,X23) = aa(nat,nat,suc,Y23) )
    <=> ( X23 = Y23 ) ) ).

% nat.inject
tff(fact_767_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_768_Int__iff,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
    <=> ( member(A,C3,A4)
        & member(A,C3,B2) ) ) ).

% Int_iff
tff(fact_769_IntI,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,A4)
     => ( member(A,C3,B2)
       => member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) ) ) ).

% IntI
tff(fact_770_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : divide_divide(A,A3,zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_771_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : divide_divide(A,zero_zero(A),A3) = zero_zero(A) ) ).

% div_0
tff(fact_772_Suc__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_less_eq
tff(fact_773_Suc__mono,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) ) ).

% Suc_mono
tff(fact_774_lessI,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,Na)) ).

% lessI
tff(fact_775_Suc__le__mono,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ).

% Suc_le_mono
tff(fact_776_diff__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% diff_Suc_Suc
tff(fact_777_Suc__diff__diff,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na)),aa(nat,nat,suc,K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),Na)),K) ).

% Suc_diff_diff
tff(fact_778_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ).

% div_self
tff(fact_779_zero__less__Suc,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,Na)) ).

% zero_less_Suc
tff(fact_780_less__Suc0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Na = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_781_diff__Suc__1,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,Na)),one_one(nat)) = Na ).

% diff_Suc_1
tff(fact_782_zle__diff1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),aa(int,int,minus_minus(int,Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ).

% zle_diff1_eq
tff(fact_783_Suc__pred,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% Suc_pred
tff(fact_784_card__insert__disjoint,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ~ member(A,Xa,A4)
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A4)) ) ) ) ).

% card_insert_disjoint
tff(fact_785_enumerate__mono__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),M: nat,Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S),M)),aa(nat,A,infini527867602293511546merate(A,S),Na))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% enumerate_mono_iff
tff(fact_786_Suc__diff__1,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) = Na ) ) ).

% Suc_diff_1
tff(fact_787_int__ops_I6_J,axiom,
    ! [A3: nat,B3: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,A3),B3)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)),zero_zero(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3))) ).

% int_ops(6)
tff(fact_788_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K)
     => ( aa(int,$o,P,aa(int,int,minus_minus(int,K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_789_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_790_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),K)),J) ).

% diff_commute
tff(fact_791_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N2: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N2))
           => aa(nat,$o,P,N2) )
       => aa(nat,$o,P,aa(nat,nat,minus_minus(nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_792_Int__left__commute,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C2)) ).

% Int_left_commute
tff(fact_793_Int__left__absorb,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) ).

% Int_left_absorb
tff(fact_794_Int__commute,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),A4) ).

% Int_commute
tff(fact_795_Int__absorb,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),A4) = A4 ).

% Int_absorb
tff(fact_796_Int__assoc,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),C2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),C2)) ).

% Int_assoc
tff(fact_797_IntD2,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
     => member(A,C3,B2) ) ).

% IntD2
tff(fact_798_IntD1,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
     => member(A,C3,A4) ) ).

% IntD1
tff(fact_799_IntE,axiom,
    ! [A: $tType,C3: A,A4: set(A),B2: set(A)] :
      ( member(A,C3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
     => ~ ( member(A,C3,A4)
         => ~ member(A,C3,B2) ) ) ).

% IntE
tff(fact_800_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_801_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_802_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_803_less__int__code_I1_J,axiom,
    ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_804_verit__la__generic,axiom,
    ! [A3: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),Xa)
      | ( A3 = Xa )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),A3) ) ).

% verit_la_generic
tff(fact_805_less__eq__int__code_I1_J,axiom,
    aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_806_Suc__inject,axiom,
    ! [Xa: nat,Ya: nat] :
      ( ( aa(nat,nat,suc,Xa) = aa(nat,nat,suc,Ya) )
     => ( Xa = Ya ) ) ).

% Suc_inject
tff(fact_807_n__not__Suc__n,axiom,
    ! [Na: nat] : Na != aa(nat,nat,suc,Na) ).

% n_not_Suc_n
tff(fact_808_imp__le__cong,axiom,
    ! [Xa: int,X5: int,P: $o,P2: $o] :
      ( ( Xa = X5 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
           => (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
           => (P2) ) ) ) ) ).

% imp_le_cong
tff(fact_809_conj__le__cong,axiom,
    ! [Xa: int,X5: int,P: $o,P2: $o] :
      ( ( Xa = X5 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
         => ( (P)
          <=> (P2) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
            & (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X5)
            & (P2) ) ) ) ) ).

% conj_le_cong
tff(fact_810_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_12: nat] : aa(nat,$o,P,X_12)
       => ? [N2: nat] :
            ( ~ aa(nat,$o,P,N2)
            & aa(nat,$o,P,aa(nat,nat,suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_811_enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S),Na)),aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,Na))) ) ) ).

% enumerate_step
tff(fact_812_vebt__buildup_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero(nat) )
     => ( ( Xa != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va: nat] : Xa != aa(nat,nat,suc,aa(nat,nat,suc,Va)) ) ) ).

% vebt_buildup.cases
tff(fact_813_not0__implies__Suc,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ? [M4: nat] : Na = aa(nat,nat,suc,M4) ) ).

% not0_implies_Suc
tff(fact_814_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_815_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_816_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_817_zero__induct,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N2: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N2))
           => aa(nat,$o,P,N2) )
       => aa(nat,$o,P,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_818_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [X3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X3),zero_zero(nat))
     => ( ! [Y: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y))
       => ( ! [X3: nat,Y: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X3),Y)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ) ).

% diff_induct
tff(fact_819_nat__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,P,N2)
           => aa(nat,$o,P,aa(nat,nat,suc,N2)) )
       => aa(nat,$o,P,Na) ) ) ).

% nat_induct
tff(fact_820_old_Onat_Oexhaust,axiom,
    ! [Ya: nat] :
      ( ( Ya != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Ya != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_821_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa(nat,nat,suc,X23) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_822_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_823_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_824_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero(nat) != aa(nat,nat,suc,X23) ).

% nat.distinct(1)
tff(fact_825_not__less__less__Suc__eq,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_826_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I2))
               => aa(nat,$o,P,I2) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_827_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_828_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_829_Suc__less__SucD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_less_SucD
tff(fact_830_less__antisym,axiom,
    ! [Na: nat,M: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
       => ( M = Na ) ) ) ).

% less_antisym
tff(fact_831_Suc__less__eq2,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M)
    <=> ? [M6: nat] :
          ( ( M = aa(nat,nat,suc,M6) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M6) ) ) ).

% Suc_less_eq2
tff(fact_832_All__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Na))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Na)
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
           => aa(nat,$o,P,I4) ) ) ) ).

% All_less_Suc
tff(fact_833_not__less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M)) ) ).

% not_less_eq
tff(fact_834_less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( M = Na ) ) ) ).

% less_Suc_eq
tff(fact_835_Ex__less__Suc,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Na))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,Na)
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
            & aa(nat,$o,P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_836_less__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na)) ) ).

% less_SucI
tff(fact_837_less__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ( M = Na ) ) ) ).

% less_SucE
tff(fact_838_Suc__lessI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( ( aa(nat,nat,suc,M) != Na )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Na) ) ) ).

% Suc_lessI
tff(fact_839_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_840_Suc__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_lessD
tff(fact_841_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_842_diff__less__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,M),Na)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_843_Suc__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na))) = aa(nat,nat,minus_minus(nat,M),Na) ) ) ).

% Suc_diff_Suc
tff(fact_844_transitive__stepwise__le,axiom,
    ! [M: nat,Na: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( ! [X3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X3),X3)
       => ( ! [X3: nat,Y: nat,Z2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X3),Y)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y),Z2)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X3),Z2) ) )
         => ( ! [N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N2),aa(nat,nat,suc,N2))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,M),Na) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_845_nat__induct__at__least,axiom,
    ! [M: nat,Na: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,P,M)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
             => ( aa(nat,$o,P,N2)
               => aa(nat,$o,P,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_induct_at_least
tff(fact_846_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M3)),N2)
             => aa(nat,$o,P,M3) )
         => aa(nat,$o,P,N2) )
     => aa(nat,$o,P,Na) ) ).

% full_nat_induct
tff(fact_847_not__less__eq__eq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M) ) ).

% not_less_eq_eq
tff(fact_848_Suc__n__not__le__n,axiom,
    ! [Na: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),Na) ).

% Suc_n_not_le_n
tff(fact_849_le__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        | ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_Suc_eq
tff(fact_850_Suc__le__D,axiom,
    ! [Na: nat,M7: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M7)
     => ? [M4: nat] : M7 = aa(nat,nat,suc,M4) ) ).

% Suc_le_D
tff(fact_851_le__SucI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na)) ) ).

% le_SucI
tff(fact_852_le__SucE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( M = aa(nat,nat,suc,Na) ) ) ) ).

% le_SucE
tff(fact_853_Suc__leD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% Suc_leD
tff(fact_854_Suc__diff__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% Suc_diff_le
tff(fact_855_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na) ).

% diff_Suc_eq_diff_pred
tff(fact_856_enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => member(A,aa(nat,A,infini527867602293511546merate(A,S),Na),S) ) ) ).

% enumerate_in_set
tff(fact_857_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E2)
         => ~ ! [N2: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),E2) ) ) ).

% nat_approx_posE
tff(fact_858_enumerate__Ex,axiom,
    ! [S: set(nat),S3: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => ( member(nat,S3,S)
       => ? [N2: nat] : aa(nat,nat,infini527867602293511546merate(nat,S),N2) = S3 ) ) ).

% enumerate_Ex
tff(fact_859_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,M: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Na)),aa(nat,A,F2,M))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_860_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N5: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,Na)),aa(nat,A,F2,N5)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_861_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Na)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_862_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N5: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N2))),aa(nat,A,F2,N2))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N5)),aa(nat,A,F2,Na)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_863_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),Na: nat,N5: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,F2,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N5)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,Na)),aa(nat,A,F2,N5)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_864_finite__enumerate__step,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),S))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S),Na)),aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,Na))) ) ) ) ).

% finite_enumerate_step
tff(fact_865_enumerate__Suc_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat] : aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,Na)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,infini527867602293511546merate(A,S),zero_zero(nat))),bot_bot(set(A))))),Na) ) ).

% enumerate_Suc'
tff(fact_866_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_867_gr0__implies__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ? [M4: nat] : Na = aa(nat,nat,suc,M4) ) ).

% gr0_implies_Suc
tff(fact_868_All__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Na))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_869_gr0__conv__Suc,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
    <=> ? [M2: nat] : Na = aa(nat,nat,suc,M2) ) ).

% gr0_conv_Suc
tff(fact_870_Ex__less__Suc2,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,suc,Na))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
            & aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_871_diff__Suc__less,axiom,
    ! [Na: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,I))),Na) ) ).

% diff_Suc_less
tff(fact_872_le__imp__less__Suc,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na)) ) ).

% le_imp_less_Suc
tff(fact_873_less__eq__Suc__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),M) ) ).

% less_eq_Suc_le
tff(fact_874_less__Suc__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_Suc_eq_le
tff(fact_875_le__less__Suc__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
      <=> ( Na = M ) ) ) ).

% le_less_Suc_eq
tff(fact_876_Suc__le__lessD,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_le_lessD
tff(fact_877_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),J)
               => ( aa(nat,$o,P,aa(nat,nat,suc,N2))
                 => aa(nat,$o,P,N2) ) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% inc_induct
tff(fact_878_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),J)
               => ( aa(nat,$o,P,N2)
                 => aa(nat,$o,P,aa(nat,nat,suc,N2)) ) ) )
         => aa(nat,$o,P,J) ) ) ) ).

% dec_induct
tff(fact_879_Suc__le__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_le_eq
tff(fact_880_Suc__leI,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na) ) ).

% Suc_leI
tff(fact_881_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_882_le__enumerate,axiom,
    ! [S: set(nat),Na: nat] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,infini527867602293511546merate(nat,S),Na)) ) ).

% le_enumerate
tff(fact_883_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),Na)
            & ! [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),K2)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,aa(nat,nat,suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_884_nat__induct__non__zero,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => ( aa(nat,$o,P,N2)
               => aa(nat,$o,P,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_induct_non_zero
tff(fact_885_Suc__diff__eq__diff__pred,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_886_Suc__pred_H,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( Na = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_887_card__insert__if,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = $ite(member(A,Xa,A4),aa(set(A),nat,finite_card(A),A4),aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A4))) ) ) ).

% card_insert_if
tff(fact_888_card__Suc__eq__finite,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B11: set(A)] :
          ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B7),B11) )
          & ~ member(A,B7,B11)
          & ( aa(set(A),nat,finite_card(A),B11) = K )
          & aa(set(A),$o,finite_finite2(A),B11) ) ) ).

% card_Suc_eq_finite
tff(fact_889_zle__int,axiom,
    ! [M: nat,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% zle_int
tff(fact_890_invar__vebt_Ointros_I1_J,axiom,
    ! [A3: $o,B3: $o] : vEBT_invar_vebt(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_891_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_892_enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,Na: nat,S: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( ~ aa(set(A),$o,finite_finite2(A),S)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S),M)),aa(nat,A,infini527867602293511546merate(A,S),Na)) ) ) ) ).

% enumerate_mono
tff(fact_893_finite__le__enumerate,axiom,
    ! [S: set(nat),Na: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(nat),nat,finite_card(nat),S))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,infini527867602293511546merate(nat,S),Na)) ) ) ).

% finite_le_enumerate
tff(fact_894_finite__enumerate__in__set,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S))
           => member(A,aa(nat,A,infini527867602293511546merate(A,S),Na),S) ) ) ) ).

% finite_enumerate_in_set
tff(fact_895_finite__enumerate__Ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),S3: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( member(A,S3,S)
           => ? [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(set(A),nat,finite_card(A),S))
                & ( aa(nat,A,infini527867602293511546merate(A,S),N2) = S3 ) ) ) ) ) ).

% finite_enumerate_Ex
tff(fact_896_finite__enum__ext,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [X4: set(A),Y3: set(A)] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(set(A),nat,finite_card(A),X4))
             => ( aa(nat,A,infini527867602293511546merate(A,X4),I2) = aa(nat,A,infini527867602293511546merate(A,Y3),I2) ) )
         => ( aa(set(A),$o,finite_finite2(A),X4)
           => ( aa(set(A),$o,finite_finite2(A),Y3)
             => ( ( aa(set(A),nat,finite_card(A),X4) = aa(set(A),nat,finite_card(A),Y3) )
               => ( X4 = Y3 ) ) ) ) ) ) ).

% finite_enum_ext
tff(fact_897_card__Suc__eq,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,K) )
    <=> ? [B7: A,B11: set(A)] :
          ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B7),B11) )
          & ~ member(A,B7,B11)
          & ( aa(set(A),nat,finite_card(A),B11) = K )
          & ( ( K = zero_zero(nat) )
           => ( B11 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_898_card__eq__SucD,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,K) )
     => ? [B5: A,B4: set(A)] :
          ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B5),B4) )
          & ~ member(A,B5,B4)
          & ( aa(set(A),nat,finite_card(A),B4) = K )
          & ( ( K = zero_zero(nat) )
           => ( B4 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_899_card__1__singleton__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A4) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ? [X: A] : A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_900_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(nat,nat,suc,zero_zero(nat)))
      <=> ! [X: A] :
            ( member(A,X,A4)
           => ! [Xa3: A] :
                ( member(A,Xa3,A4)
               => ( X = Xa3 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_901_card__le__Suc__iff,axiom,
    ! [A: $tType,Na: nat,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),A4))
    <=> ? [A9: A,B11: set(A)] :
          ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A9),B11) )
          & ~ member(A,A9,B11)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(set(A),nat,finite_card(A),B11))
          & aa(set(A),$o,finite_finite2(A),B11) ) ) ).

% card_le_Suc_iff
tff(fact_902_finite__enumerate__mono,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M: nat,Na: nat,S: set(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(A),$o,finite_finite2(A),S)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),S))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,S),M)),aa(nat,A,infini527867602293511546merate(A,S),Na)) ) ) ) ) ).

% finite_enumerate_mono
tff(fact_903_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_904_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_905_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_906_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_907_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% zero_less_divide_1_iff
tff(fact_908_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_909_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_910_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_911_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A3)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_912_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,one_one(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_913_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( divide_divide(A,A3,B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_914_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( divide_divide(A,C3,A3) = divide_divide(A,C3,B3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% divide_cancel_left
tff(fact_915_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,C3: A,B3: A] :
          ( ( divide_divide(A,A3,C3) = divide_divide(A,B3,C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% divide_cancel_right
tff(fact_916_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : divide_divide(A,A3,zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_917_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( divide_divide(A,A3,B3) = one_one(A) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_918_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( one_one(A) = divide_divide(A,A3,B3) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_919_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,A3,A3) = one_one(A) ) ) ) ).

% divide_self
tff(fact_920_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          divide_divide(A,A3,A3) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_921_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( divide_divide(A,B3,A3) = one_one(A) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_922_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( one_one(A) = divide_divide(A,B3,A3) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_923_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( divide_divide(A,one_one(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_924_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = divide_divide(A,one_one(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_925_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% zero_le_divide_1_iff
tff(fact_926_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_927_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X2: A] :
        ? [Y: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X2) ) ).

% linordered_field_no_lb
tff(fact_928_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X2: A] :
        ? [X_1: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),X_1) ) ).

% linordered_field_no_ub
tff(fact_929_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C3)),divide_divide(A,A3,C3)) ) ) ) ).

% divide_right_mono_neg
tff(fact_930_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xa,Ya)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_931_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Ya)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_932_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Ya)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_933_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xa,Ya)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_934_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,A3,B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_935_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A3,C3)),divide_divide(A,B3,C3)) ) ) ) ).

% divide_right_mono
tff(fact_936_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A3,B3)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ) ) ).

% divide_le_0_iff
tff(fact_937_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,Xa,Ya)) ) ) ) ).

% divide_neg_neg
tff(fact_938_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Ya)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_939_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Ya)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_940_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,Xa,Ya)) ) ) ) ).

% divide_pos_pos
tff(fact_941_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A3,B3)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ) ) ).

% divide_less_0_iff
tff(fact_942_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A3,C3)),divide_divide(A,B3,C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) )
            & ( C3 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_943_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A3,B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_944_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A3,C3)),divide_divide(A,B3,C3)) ) ) ) ).

% divide_strict_right_mono
tff(fact_945_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,A3,C3)),divide_divide(A,B3,C3)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_946_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( divide_divide(A,A3,B3) = one_one(A) )
          <=> ( A3 = B3 ) ) ) ) ).

% right_inverse_eq
tff(fact_947_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Ya)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_948_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xa,Ya)) ) ) ) ).

% divide_nonpos_neg
tff(fact_949_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,Xa,Ya)) ) ) ) ).

% divide_nonneg_pos
tff(fact_950_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Ya)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_951_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A3,C3)),divide_divide(A,B3,C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% divide_le_cancel
tff(fact_952_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Z)),divide_divide(A,Ya,W2)) ) ) ) ) ) ).

% frac_less2
tff(fact_953_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Z)),divide_divide(A,Ya,W2)) ) ) ) ) ) ).

% frac_less
tff(fact_954_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ya: A,Xa: A,W2: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),W2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Z)),divide_divide(A,Ya,W2)) ) ) ) ) ) ).

% frac_le
tff(fact_955_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,A3)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_956_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),divide_divide(A,B3,A3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% less_divide_eq_1
tff(fact_957_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),divide_divide(A,B3,A3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% le_divide_eq_1
tff(fact_958_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,A3)),one_one(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_959_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_960_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_961_le__div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ( divide_divide(nat,M,Na) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),Na),Na)) ) ) ) ).

% le_div_geq
tff(fact_962_div__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( divide_divide(nat,M,Na) = zero_zero(nat) ) ) ).

% div_less
tff(fact_963_div__by__Suc__0,axiom,
    ! [M: nat] : divide_divide(nat,M,aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_964_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : divide_divide(A,zero_zero(A),A3) = zero_zero(A) ) ).

% bits_div_0
tff(fact_965_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : divide_divide(A,A3,zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_966_Suc__if__eq,axiom,
    ! [A: $tType,F2: fun(nat,A),H: fun(nat,A),G: A,Na: 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 )
       => ( aa(nat,A,F2,Na) = $ite(Na = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_967_real__of__nat__div3,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Na),aa(nat,real,semiring_1_of_nat(real),Xa))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Na,Xa)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_968_real__of__nat__div4,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Na,Xa))),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Na),aa(nat,real,semiring_1_of_nat(real),Xa))) ).

% real_of_nat_div4
tff(fact_969_real__of__nat__div2,axiom,
    ! [Na: nat,Xa: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,minus_minus(real,divide_divide(real,aa(nat,real,semiring_1_of_nat(real),Na),aa(nat,real,semiring_1_of_nat(real),Xa))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Na,Xa)))) ).

% real_of_nat_div2
tff(fact_970_div__le__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,M,Na)),M) ).

% div_le_dividend
tff(fact_971_div__le__mono,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,M,K)),divide_divide(nat,Na,K)) ) ).

% div_le_mono
tff(fact_972_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( divide_divide(nat,M,Na) = zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( Na = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_973_Suc__div__le__mono,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,M,Na)),divide_divide(nat,aa(nat,nat,suc,M),Na)) ).

% Suc_div_le_mono
tff(fact_974_div__le__mono2,axiom,
    ! [M: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,K,Na)),divide_divide(nat,K,M)) ) ) ).

% div_le_mono2
tff(fact_975_div__greater__zero__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,M,Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% div_greater_zero_iff
tff(fact_976_div__eq__dividend__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( ( divide_divide(nat,M,Na) = M )
      <=> ( Na = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_977_div__less__dividend,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,M,Na)),M) ) ) ).

% div_less_dividend
tff(fact_978_div__if,axiom,
    ! [M: nat,Na: nat] :
      divide_divide(nat,M,Na) = $ite(
        ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | ( Na = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),Na),Na)) ) ).

% div_if
tff(fact_979_div__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ( divide_divide(nat,M,Na) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),Na),Na)) ) ) ) ).

% div_geq
tff(fact_980_int__div__less__self,axiom,
    ! [Xa: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,Xa,K)),Xa) ) ) ).

% int_div_less_self
tff(fact_981_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),divide_divide(int,A3,B3))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),A3)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_982_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,A3,B3))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_983_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,A3,B3))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_984_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),divide_divide(int,I,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_985_div__nonpos__pos__le0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A3,B3)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_986_div__nonneg__neg__le0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A3,B3)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_987_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),divide_divide(int,K,L)) ) ) ).

% div_positive_int
tff(fact_988_ln__inj__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
       => ( ( aa(real,real,ln_ln(real),Xa) = aa(real,real,ln_ln(real),Ya) )
        <=> ( Xa = Ya ) ) ) ) ).

% ln_inj_iff
tff(fact_989_ln__less__cancel__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ) ) ).

% ln_less_cancel_iff
tff(fact_990_ln__le__cancel__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ) ) ).

% ln_le_cancel_iff
tff(fact_991_ln__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_992_ln__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ).

% ln_gt_zero_iff
tff(fact_993_ln__eq__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,ln_ln(real),Xa) = zero_zero(real) )
      <=> ( Xa = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_994_ln__ge__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ).

% ln_ge_zero_iff
tff(fact_995_ln__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_996_ln__div,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
       => ( aa(real,real,ln_ln(real),divide_divide(real,Xa,Ya)) = aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Ya)) ) ) ) ).

% ln_div
tff(fact_997_ln__diff__le,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Ya))),divide_divide(real,aa(real,real,minus_minus(real,Xa),Ya),Ya)) ) ) ).

% ln_diff_le
tff(fact_998_ln__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),Xa) ) ).

% ln_bound
tff(fact_999_ln__less__self,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),Xa) ) ).

% ln_less_self
tff(fact_1000_less__eq__real__def,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% less_eq_real_def
tff(fact_1001_complete__real,axiom,
    ! [S: set(real)] :
      ( ? [X2: real] : member(real,X2,S)
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( member(real,X3,S)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),Z3) )
       => ? [Y: real] :
            ( ! [X2: real] :
                ( member(real,X2,S)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Y) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( member(real,X3,S)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),Z3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Z3) ) ) ) ) ).

% complete_real
tff(fact_1002_ln__gt__zero__imp__gt__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_1003_ln__eq__minus__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,ln_ln(real),Xa) = aa(real,real,minus_minus(real,Xa),one_one(real)) )
       => ( Xa = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_1004_ln__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),Xa)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_1005_ln__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa)) ) ).

% ln_gt_zero
tff(fact_1006_ln__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa)) ) ).

% ln_ge_zero
tff(fact_1007_ln__le__minus__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,minus_minus(real,Xa),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_1008_ln__ge__zero__imp__ge__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xa))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_1009_pos__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A3,B3)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1010_neg__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A3,B3)),zero_zero(int))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A3) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1011_div__neg__pos__less0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A3,B3)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_1012_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A3,B3)) ) ) ) ).

% div_positive
tff(fact_1013_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( divide_divide(A,A3,B3) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_1014_zdiv__mono1,axiom,
    ! [A3: int,A11: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),A11)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A3,B3)),divide_divide(int,A11,B3)) ) ) ).

% zdiv_mono1
tff(fact_1015_zdiv__mono2,axiom,
    ! [A3: int,B9: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B9)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B9),B3)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A3,B3)),divide_divide(int,A3,B9)) ) ) ) ).

% zdiv_mono2
tff(fact_1016_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( divide_divide(int,I,K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1017_zdiv__mono1__neg,axiom,
    ! [A3: int,A11: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),A11)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A11,B3)),divide_divide(int,A3,B3)) ) ) ).

% zdiv_mono1_neg
tff(fact_1018_zdiv__mono2__neg,axiom,
    ! [A3: int,B9: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B9)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B9),B3)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,A3,B9)),divide_divide(int,A3,B3)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1019_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,K,L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_1020_Bolzano,axiom,
    ! [A3: real,B3: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
     => ( ! [A5: real,B5: real,C5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A5),B5)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B5),C5)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A5),B5)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B5),C5)
                 => aa(real,$o,aa(real,fun(real,$o),P,A5),C5) ) ) ) )
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
               => ? [D4: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                    & ! [A5: real,B5: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A5),X3)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B5)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,minus_minus(real,B5),A5)),D4) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A5),B5) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A3),B3) ) ) ) ).

% Bolzano
tff(fact_1021_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => ( divide_divide(int,aa(nat,int,power_power(int,K),M),K) = aa(nat,int,power_power(int,K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1022_nat__ivt__aux,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_1023_list__decode_Ocases,axiom,
    ! [Xa: nat] :
      ( ( Xa != zero_zero(nat) )
     => ~ ! [N2: nat] : Xa != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_1024_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_12)
     => ( ! [X3: A,N2: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N2),X3)
           => ? [Y2: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N2)),Y2)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N2),X3),Y2) ) )
       => ? [F5: fun(nat,A)] :
          ! [N3: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N3),aa(nat,A,F5,N3))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N3),aa(nat,A,F5,N3)),aa(nat,A,F5,aa(nat,nat,suc,N3))) ) ) ) ).

% dependent_nat_choice
tff(fact_1025_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,aa(int,int,minus_minus(int,K),L),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1026_nat__intermed__int__val,axiom,
    ! [M: nat,Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,suc,I2))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,M)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
           => ? [I2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I2)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
                & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_1027_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_1028_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
        <=> ( B3 = C3 ) ) ) ).

% add_left_cancel
tff(fact_1029_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B3: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
        <=> ( B3 = C3 ) ) ) ).

% add_right_cancel
tff(fact_1030_abs__idempotent,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_idempotent
tff(fact_1031_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% add_le_cancel_right
tff(fact_1032_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% add_le_cancel_left
tff(fact_1033_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_1034_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add_0
tff(fact_1035_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Ya: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Ya = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_1036_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Ya = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_1037_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_1038_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_1039_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = A3 )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_1040_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = A3 )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_1041_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_1042_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.right_neutral
tff(fact_1043_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% add_less_cancel_left
tff(fact_1044_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% add_less_cancel_right
tff(fact_1045_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3) = A3 ) ).

% add_diff_cancel
tff(fact_1046_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A3),B3)),B3) = A3 ) ).

% diff_add_cancel
tff(fact_1047_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C3: A,A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3)) = aa(A,A,minus_minus(A,A3),B3) ) ).

% add_diff_cancel_left
tff(fact_1048_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),A3) = B3 ) ).

% add_diff_cancel_left'
tff(fact_1049_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,C3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) = aa(A,A,minus_minus(A,A3),B3) ) ).

% add_diff_cancel_right
tff(fact_1050_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3) = A3 ) ).

% add_diff_cancel_right'
tff(fact_1051_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_1052_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_1053_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,abs_abs(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_1054_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_1055_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = 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),Na)) ) ).

% of_nat_add
tff(fact_1056_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)) ) ).

% abs_add_abs
tff(fact_1057_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),Na)) = aa(nat,A,semiring_1_of_nat(A),Na) ) ).

% abs_of_nat
tff(fact_1058_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_1059_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_1060_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ).

% le_add_same_cancel1
tff(fact_1061_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ).

% le_add_same_cancel2
tff(fact_1062_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1063_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1064_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_1065_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_1066_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ).

% less_add_same_cancel1
tff(fact_1067_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ).

% less_add_same_cancel2
tff(fact_1068_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1069_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1070_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,minus_minus(A,A3),B3)) = A3 ) ) ) ).

% le_add_diff_inverse
tff(fact_1071_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A3),B3)),B3) = A3 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1072_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_1073_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_1074_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% abs_le_self_iff
tff(fact_1075_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_1076_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A3))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_1077_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,A3,aa(A,A,abs_abs(A),B3))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_1078_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),divide_divide(A,A3,aa(A,A,abs_abs(A),B3)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            | ( B3 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_1079_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_1080_nat__1,axiom,
    aa(int,nat,nat2,one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_1081_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_1082_nat__0__iff,axiom,
    ! [I: int] :
      ( ( aa(int,nat,nat2,I) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_1083_zless__nat__conj,axiom,
    ! [W2: int,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% zless_nat_conj
tff(fact_1084_zle__add1__eq__le,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ).

% zle_add1_eq_le
tff(fact_1085_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),aa(int,nat,nat2,Z)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_1086_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_1087_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_1088_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))) ) ).

% abs_triangle_ineq
tff(fact_1089_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) ) ).

% is_num_normalize(1)
tff(fact_1090_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) ) ).

% ab_semigroup_add_class.add_ac(1)
tff(fact_1091_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_1092_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: A,K: A,A3: A,B3: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% group_cancel.add1
tff(fact_1093_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B2: A,K: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B3) )
         => ( 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),A3),B3)) ) ) ) ).

% group_cancel.add2
tff(fact_1094_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) ) ).

% add.assoc
tff(fact_1095_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
        <=> ( B3 = C3 ) ) ) ).

% add.left_cancel
tff(fact_1096_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B3: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
        <=> ( B3 = C3 ) ) ) ).

% add.right_cancel
tff(fact_1097_add_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) ) ).

% add.commute
tff(fact_1098_add_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B3: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) ) ).

% add.left_commute
tff(fact_1099_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
         => ( B3 = C3 ) ) ) ).

% add_left_imp_eq
tff(fact_1100_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B3: A,A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) )
         => ( B3 = C3 ) ) ) ).

% add_right_imp_eq
tff(fact_1101_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A3: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xa),A3))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A3),R2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_1102_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A,C3: A,D2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),C3))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B3),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_1103_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),B3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))) ) ).

% abs_triangle_ineq4
tff(fact_1104_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,A3: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xa),A3))),R2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A3),R2)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_1105_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_self
tff(fact_1106_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% abs_le_D1
tff(fact_1107_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_1108_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),B3)) = aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B3),A3)) ) ).

% abs_minus_commute
tff(fact_1109_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% add_le_imp_le_right
tff(fact_1110_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% add_le_imp_le_left
tff(fact_1111_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ? [C6: A] : B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C6) ) ) ).

% le_iff_add
tff(fact_1112_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) ) ) ).

% add_right_mono
tff(fact_1113_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ~ ! [C5: A] : B3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C5) ) ) ).

% less_eqE
tff(fact_1114_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3)) ) ) ).

% add_left_mono
tff(fact_1115_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D2)) ) ) ) ).

% add_mono
tff(fact_1116_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_1117_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_1118_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_1119_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add.group_left_neutral
tff(fact_1120_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.comm_neutral
tff(fact_1121_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% comm_monoid_add_class.add_0
tff(fact_1122_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% verit_sum_simplify
tff(fact_1123_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_1124_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_1125_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_1126_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D2)) ) ) ) ).

% add_strict_mono
tff(fact_1127_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3)) ) ) ).

% add_strict_left_mono
tff(fact_1128_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) ) ) ).

% add_strict_right_mono
tff(fact_1129_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% add_less_imp_less_left
tff(fact_1130_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% add_less_imp_less_right
tff(fact_1131_infinite__int__iff__unbounded__le,axiom,
    ! [S: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S)
    <=> ! [M2: int] :
        ? [N: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M2),aa(int,int,abs_abs(int),N))
          & member(int,N,S) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_1132_infinite__int__iff__unbounded,axiom,
    ! [S: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S)
    <=> ! [M2: int] :
        ? [N: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M2),aa(int,int,abs_abs(int),N))
          & member(int,N,S) ) ) ).

% infinite_int_iff_unbounded
tff(fact_1133_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A4: A,K: A,A3: A,B3: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,minus_minus(A,A4),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,minus_minus(A,A3),B3)) ) ) ) ).

% group_cancel.sub1
tff(fact_1134_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = C3 )
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3) ) ) ) ).

% diff_eq_eq
tff(fact_1135_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,C3: A,B3: A] :
          ( ( A3 = aa(A,A,minus_minus(A,C3),B3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = C3 ) ) ) ).

% eq_diff_eq
tff(fact_1136_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,minus_minus(A,B3),C3)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C3) ) ).

% add_diff_eq
tff(fact_1137_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,minus_minus(A,B3),C3)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B3) ) ).

% diff_diff_eq2
tff(fact_1138_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A3),B3)),C3) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B3) ) ).

% diff_add_eq
tff(fact_1139_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A3),C3)),B3) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1140_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C3: A,B3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3) = A3 )
         => ( C3 = aa(A,A,minus_minus(A,A3),B3) ) ) ) ).

% add_implies_diff
tff(fact_1141_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A3),B3)),C3) = aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) ) ).

% diff_diff_eq
tff(fact_1142_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),Xa))) ) ).

% abs_add_one_gt_zero
tff(fact_1143_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_zero
tff(fact_1144_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,A,abs_abs(A),A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_1145_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_1146_nat__abs__int__diff,axiom,
    ! [A3: nat,B3: nat] :
      aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3),aa(nat,nat,minus_minus(nat,B3),A3),aa(nat,nat,minus_minus(nat,A3),B3)) ).

% nat_abs_int_diff
tff(fact_1147_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),B3))) ) ).

% abs_triangle_ineq2
tff(fact_1148_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A3),B3))) ) ).

% abs_triangle_ineq3
tff(fact_1149_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B3),A3))) ) ).

% abs_triangle_ineq2_sym
tff(fact_1150_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),divide_divide(A,A3,B3)) = divide_divide(A,aa(A,A,abs_abs(A),A3),aa(A,A,abs_abs(A),B3)) ) ) ) ).

% nonzero_abs_divide
tff(fact_1151_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(nat,nat,suc,aa(int,nat,nat2,Z)) = aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_1152_nat__zero__as__int,axiom,
    zero_zero(nat) = aa(int,nat,nat2,zero_zero(int)) ).

% nat_zero_as_int
tff(fact_1153_nat__mono,axiom,
    ! [Xa: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Ya)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Ya)) ) ).

% nat_mono
tff(fact_1154_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
       => ( ( aa(int,nat,nat2,Z) = aa(int,nat,nat2,Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_1155_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_13: nat] : aa(nat,$o,P,X_13)
    <=> ! [X: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
         => aa(nat,$o,P,aa(int,nat,nat2,X)) ) ) ).

% all_nat
tff(fact_1156_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_13: nat] : aa(nat,$o,P,X_13)
    <=> ? [X: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
          & aa(nat,$o,P,aa(int,nat,nat2,X)) ) ) ).

% ex_nat
tff(fact_1157_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B3) ) ) ) ).

% add_decreasing
tff(fact_1158_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) ) ) ) ).

% add_increasing
tff(fact_1159_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),B3) ) ) ) ).

% add_decreasing2
tff(fact_1160_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) ) ) ) ).

% add_increasing2
tff(fact_1161_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_1162_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_1163_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Ya = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_1164_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya) = zero_zero(A) )
            <=> ( ( Xa = zero_zero(A) )
                & ( Ya = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_1165_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_1166_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_1167_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_1168_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_1169_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_1170_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_pos_pos
tff(fact_1171_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ ! [C5: A] :
                ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C5) )
               => ( C5 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1172_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) ) ) ) ).

% pos_add_strict
tff(fact_1173_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_1174_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => ( ( aa(A,A,minus_minus(A,B3),A3) = C3 )
            <=> ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1175_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,minus_minus(A,B3),A3)) = B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1176_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,minus_minus(A,C3),aa(A,A,minus_minus(A,B3),A3)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),B3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1177_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B3),A3)),C3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1178_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B3),A3)),C3) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1179_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,minus_minus(A,B3),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1180_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),aa(A,A,minus_minus(A,B3),A3)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1181_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,minus_minus(A,B3),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),A3)),B3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1182_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)),A3)) ) ) ).

% le_add_diff
tff(fact_1183_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B3),A3)),A3) = B3 ) ) ) ).

% diff_add
tff(fact_1184_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,minus_minus(A,C3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C3) ) ) ).

% le_diff_eq
tff(fact_1185_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A3),B3)),C3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3)) ) ) ).

% diff_le_eq
tff(fact_1186_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,minus_minus(A,Na),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_1187_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Na: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Na),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Na),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1188_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).

% less_add_one
tff(fact_1189_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),one_one(A))) ) ) ).

% add_mono1
tff(fact_1190_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,A3),B3)),C3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),B3)) ) ) ).

% diff_less_eq
tff(fact_1191_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,minus_minus(A,C3),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C3) ) ) ).

% less_diff_eq
tff(fact_1192_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,minus_minus(A,A3),B3)) = A3 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1193_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_1194_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_1195_zless__add1__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
        | ( W2 = Z ) ) ) ).

% zless_add1_eq
tff(fact_1196_zle__iff__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z)
    <=> ? [N: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% zle_iff_zadd
tff(fact_1197_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xa: A] : neg_numeral_dbl_inc(A,Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Xa)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_1198_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xa: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),E) )
         => ( Xa = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_1199_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
         => ( divide_divide(A,aa(A,A,abs_abs(A),Xa),Ya) = aa(A,A,abs_abs(A),divide_divide(A,Xa,Ya)) ) ) ) ).

% abs_div_pos
tff(fact_1200_nat__mono__iff,axiom,
    ! [Z: int,W2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% nat_mono_iff
tff(fact_1201_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(int,nat,nat2,Z))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_1202_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1203_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_nonneg_pos
tff(fact_1204_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1205_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_pos_nonneg
tff(fact_1206_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) ) ) ) ).

% add_strict_increasing
tff(fact_1207_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3)) ) ) ) ).

% add_strict_increasing2
tff(fact_1208_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( ! [E: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),E)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ya),E)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ).

% field_le_epsilon
tff(fact_1209_nat__le__iff,axiom,
    ! [Xa: int,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,Xa)),Na)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(nat,int,semiring_1_of_nat(int),Na)) ) ).

% nat_le_iff
tff(fact_1210_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_1211_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = aa(int,nat,nat2,Z) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_1212_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B3) ) ) ).

% discrete
tff(fact_1213_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))) ) ).

% zero_less_two
tff(fact_1214_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B3)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_1215_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,B3)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_1216_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B3) ) ) ).

% gt_half_sum
tff(fact_1217_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_1218_zless__iff__Suc__zadd,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
    <=> ? [N: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N))) ) ).

% zless_iff_Suc_zadd
tff(fact_1219_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_1220_add1__zle__eq,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ).

% add1_zle_eq
tff(fact_1221_zless__imp__add1__zle,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W2),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_1222_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I2: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_1223_nat__less__eq__zless,axiom,
    ! [W2: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% nat_less_eq_zless
tff(fact_1224_nat__le__eq__zle,axiom,
    ! [W2: int,Z: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),W2)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,W2)),aa(int,nat,nat2,Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ) ).

% nat_le_eq_zle
tff(fact_1225_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M = aa(int,nat,nat2,W2) )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2),W2 = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_1226_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( aa(int,nat,nat2,W2) = M )
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2),W2 = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_1227_split__nat,axiom,
    ! [P: fun(nat,$o),I: int] :
      ( aa(nat,$o,P,aa(int,nat,nat2,I))
    <=> ( ! [N: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N) )
           => aa(nat,$o,P,N) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_1228_le__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Na)),K) ) ) ).

% le_nat_iff
tff(fact_1229_nat__diff__distrib_H,axiom,
    ! [Xa: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
       => ( aa(int,nat,nat2,aa(int,int,minus_minus(int,Xa),Ya)) = aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,Xa)),aa(int,nat,nat2,Ya)) ) ) ) ).

% nat_diff_distrib'
tff(fact_1230_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z4)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z4),Z)
       => ( aa(int,nat,nat2,aa(int,int,minus_minus(int,Z),Z4)) = aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_1231_nat__div__distrib_H,axiom,
    ! [Ya: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
     => ( aa(int,nat,nat2,divide_divide(int,Xa,Ya)) = divide_divide(nat,aa(int,nat,nat2,Xa),aa(int,nat,nat2,Ya)) ) ) ).

% nat_div_distrib'
tff(fact_1232_nat__div__distrib,axiom,
    ! [Xa: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,nat,nat2,divide_divide(int,Xa,Ya)) = divide_divide(nat,aa(int,nat,nat2,Xa),aa(int,nat,nat2,Ya)) ) ) ).

% nat_div_distrib
tff(fact_1233_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ).

% le_imp_0_less
tff(fact_1234_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),W2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,W2)),M)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),aa(nat,int,semiring_1_of_nat(int),M)) ) ) ).

% nat_less_iff
tff(fact_1235_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,aa(nat,A,power_power(A,A3),M),aa(nat,A,power_power(A,A3),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(nat,A,power_power(A,A3),aa(nat,nat,minus_minus(nat,M),Na)),divide_divide(A,one_one(A),aa(nat,A,power_power(A,A3),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1236_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),Xa)),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
            | ( Na = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1237_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,B3),M)),aa(nat,A,power_power(A,B3),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_1238_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),Na))
        <=> ( ( A3 != zero_zero(A) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_1239_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ) ) ).

% power_mono_iff
tff(fact_1240_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,Xa: nat,Ya: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,B3),Xa)),aa(nat,A,power_power(A,B3),Ya))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),Ya) ) ) ) ).

% power_increasing_iff
tff(fact_1241_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,B3),M)),aa(nat,A,power_power(A,B3),Na))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_1242_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: nat,W2: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W2)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B3),W2)),Xa) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1243_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B3: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(nat,nat,power_power(nat,B3),W2)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1244_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: nat,W2: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W2)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B3),W2)),Xa) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1245_even__odd__cases,axiom,
    ! [Xa: nat] :
      ( ! [N2: nat] : Xa != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2)
     => ~ ! [N2: nat] : Xa != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) ) ).

% even_odd_cases
tff(fact_1246_add__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% add_Suc_right
tff(fact_1247_add__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( Na = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_1248_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_1249_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% nat_add_left_cancel_less
tff(fact_1250_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% nat_add_left_cancel_le
tff(fact_1251_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1252_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( ( aa(nat,A,power_power(A,A3),M) = aa(nat,A,power_power(A,A3),Na) )
          <=> ( M = Na ) ) ) ) ).

% power_inject_exp
tff(fact_1253_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,power_power(A,zero_zero(A)),aa(nat,nat,suc,Na)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_1254_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,power_power(A,A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% power_Suc0_right
tff(fact_1255_add__gr__0,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% add_gr_0
tff(fact_1256_power__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_1257_nat__power__eq__Suc__0__iff,axiom,
    ! [Xa: nat,M: nat] :
      ( ( aa(nat,nat,power_power(nat,Xa),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( Xa = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_1258_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) = aa(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_1259_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) = aa(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_1260_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(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_1261_nat__zero__less__power__iff,axiom,
    ! [Xa: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,power_power(nat,Xa),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Xa)
        | ( Na = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_1262_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,Xa: nat,Ya: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,B3),Xa)),aa(nat,A,power_power(A,B3),Ya))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ya) ) ) ) ).

% power_strict_increasing_iff
tff(fact_1263_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,Na: nat] :
          ( ( aa(nat,A,power_power(A,A3),Na) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% power_eq_0_iff
tff(fact_1264_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))) = aa(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_1265_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))),I) = aa(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_1266_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,B3: nat,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B3)),W2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,B3),W2)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1267_add__Suc__shift,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Na)) ).

% add_Suc_shift
tff(fact_1268_add__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% add_Suc
tff(fact_1269_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A3: nat] :
      ( ( A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A3) )
     => ( aa(nat,nat,suc,A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A3)) ) ) ).

% nat_arith.suc1
tff(fact_1270_plus__nat_Oadd__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),Na) = Na ).

% plus_nat.add_0
tff(fact_1271_add__eq__self__zero,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = M )
     => ( Na = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_1272_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),K) ) ).

% add_lessD1
tff(fact_1273_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_1274_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I) ).

% not_add_less1
tff(fact_1275_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I) ).

% not_add_less2
tff(fact_1276_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_less_mono1
tff(fact_1277_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_less_add1
tff(fact_1278_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_less_add2
tff(fact_1279_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Na) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% less_add_eq_less
tff(fact_1280_add__leE,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ) ).

% add_leE
tff(fact_1281_le__add1,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) ).

% le_add1
tff(fact_1282_le__add2,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% le_add2
tff(fact_1283_add__leD1,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% add_leD1
tff(fact_1284_add__leD2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ).

% add_leD2
tff(fact_1285_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_1286_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_1287_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_1288_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_le_add1
tff(fact_1289_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_le_add2
tff(fact_1290_nat__le__iff__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_1291_diff__add__inverse2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),Na) = M ).

% diff_add_inverse2
tff(fact_1292_diff__add__inverse,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),Na) = M ).

% diff_add_inverse
tff(fact_1293_diff__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] : aa(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),Na),K)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% diff_cancel2
tff(fact_1294_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(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),Na)) = aa(nat,nat,minus_minus(nat,M),Na) ).

% Nat.diff_cancel
tff(fact_1295_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,I),M)),aa(nat,nat,power_power(nat,I),Na))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_power_less_imp_less
tff(fact_1296_power__gt__expt,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,Na),K)) ) ).

% power_gt_expt
tff(fact_1297_nat__one__le__power,axiom,
    ! [I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,power_power(nat,I),Na)) ) ).

% nat_one_le_power
tff(fact_1298_add__is__1,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Na = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_1299_one__is__add,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Na = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_1300_real__arch__pow,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ? [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),aa(nat,real,power_power(real,Xa),N2)) ) ).

% real_arch_pow
tff(fact_1301_less__natE,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ~ ! [Q3: nat] : Na != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ) ).

% less_natE
tff(fact_1302_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M))) ).

% less_add_Suc1
tff(fact_1303_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I))) ).

% less_add_Suc2
tff(fact_1304_less__iff__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
    <=> ? [K3: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_1305_less__imp__Suc__add,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ? [K2: nat] : Na = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_1306_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ? [K2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_1307_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F2: fun(A,nat),Na: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X3: A] :
            ( aa(A,$o,P,X3)
           => ? [Y2: A] :
                ( aa(A,$o,P,Y2)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Y2)),aa(A,nat,F2,X3)) ) )
       => ? [Y: A] :
            ( aa(A,$o,P,Y)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Y)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,K)),Na)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_1308_mono__nat__linear__lb,axiom,
    ! [F2: fun(nat,nat),M: nat,K: nat] :
      ( ! [M4: nat,N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,F2,M4)),aa(nat,nat,F2,N2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F2,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_1309_diff__add__0,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1310_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,J),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_1311_add__diff__inverse__nat,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,minus_minus(nat,M),Na)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1312_Suc__eq__plus1__left,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),Na) ).

% Suc_eq_plus1_left
tff(fact_1313_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_1314_Suc__eq__plus1,axiom,
    ! [Na: nat] : aa(nat,nat,suc,Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_1315_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_1316_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,minus_minus(nat,J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_1317_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,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,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1318_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,nat,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,minus_minus(nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1319_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,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_1320_real__arch__pow__inv,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ? [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,Xa),N2)),Ya) ) ) ).

% real_arch_pow_inv
tff(fact_1321_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A3: nat,B3: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A3),B3))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D5: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),D5) )
           => aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split
tff(fact_1322_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A3: nat,B3: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A3),B3))
    <=> ~ ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D5: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),D5) )
              & ~ aa(nat,$o,P,D5) ) ) ) ).

% nat_diff_split_asm
tff(fact_1323_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_1324_card__Un__le,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2))) ).

% card_Un_le
tff(fact_1325_add__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na) = $ite(M = zero_zero(nat),Na,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na))) ).

% add_eq_if
tff(fact_1326_nat__less__real__le,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),Na)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M)) ) ).

% nat_less_real_le
tff(fact_1327_nat__le__real__less,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),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_1328_card__Un__Int,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ).

% card_Un_Int
tff(fact_1329_ln__add__one__self__le__self,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa) ) ).

% ln_add_one_self_le_self
tff(fact_1330_nat__power__eq,axiom,
    ! [Z: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(nat,int,power_power(int,Z),Na)) = aa(nat,nat,power_power(nat,aa(int,nat,nat2,Z)),Na) ) ) ).

% nat_power_eq
tff(fact_1331_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),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_1332_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(int,nat,nat2,aa(int,int,abs_abs(int),K))),aa(int,nat,nat2,aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_1333_card__Un__disjoint,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2)) ) ) ) ) ).

% card_Un_disjoint
tff(fact_1334_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,Na: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,A,power_power(A,A3),Na) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_1335_nat0__intermed__int__val,axiom,
    ! [Na: nat,F2: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,F2,I2)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F2,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F2,Na))
         => ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Na)
              & ( aa(nat,int,F2,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_1336_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% zero_le_power
tff(fact_1337_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ).

% power_mono
tff(fact_1338_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% zero_less_power
tff(fact_1339_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% one_le_power
tff(fact_1340_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A] : aa(nat,A,power_power(A,A3),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_1341_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% power_less_imp_less_base
tff(fact_1342_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),Na)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_1343_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na))),aa(nat,A,power_power(A,B3),aa(nat,nat,suc,Na)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% power_le_imp_le_base
tff(fact_1344_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( ( aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na)) = aa(nat,A,power_power(A,B3),aa(nat,nat,suc,Na)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
             => ( A3 = B3 ) ) ) ) ) ).

% power_inject_base
tff(fact_1345_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(nat,A,power_power(A,zero_zero(A)),Na) = $ite(Na = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_1346_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na))) ) ) ).

% power_gt1
tff(fact_1347_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ) ).

% power_less_imp_less_exp
tff(fact_1348_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N4: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,A3),N4)) ) ) ) ).

% power_strict_increasing
tff(fact_1349_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A3)),Na)) ) ).

% zero_le_power_abs
tff(fact_1350_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N4: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,A3),N4)) ) ) ) ).

% power_increasing
tff(fact_1351_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(nat,A,power_power(A,zero_zero(A)),Na) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_1352_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na))),A3) ) ) ) ).

% power_Suc_le_self
tff(fact_1353_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,Na))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_1354_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N4: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),N4)),aa(nat,A,power_power(A,A3),Na)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1355_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,N4: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),N4)),aa(nat,A,power_power(A,A3),Na)) ) ) ) ) ).

% power_decreasing
tff(fact_1356_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% power_le_imp_le_exp
tff(fact_1357_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( ( aa(nat,A,power_power(A,A3),Na) = aa(nat,A,power_power(A,B3),Na) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
               => ( A3 = B3 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_1358_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,A3: A,B3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
             => ( ( aa(nat,A,power_power(A,A3),Na) = aa(nat,A,power_power(A,B3),Na) )
              <=> ( A3 = B3 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_1359_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(nat,A,power_power(A,A3),Na)) ) ) ) ).

% self_le_power
tff(fact_1360_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ) ).

% one_less_power
tff(fact_1361_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,Na: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( aa(nat,A,power_power(A,A3),aa(nat,nat,minus_minus(nat,M),Na)) = divide_divide(A,aa(nat,A,power_power(A,A3),M),aa(nat,A,power_power(A,A3),Na)) ) ) ) ) ).

% power_diff
tff(fact_1362_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ) ).

% power_strict_mono
tff(fact_1363_lemma__interval,axiom,
    ! [A3: real,Xa: real,B3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B3)
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [Y2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xa),Y2))),D6)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),B3) ) ) ) ) ) ).

% lemma_interval
tff(fact_1364_realpow__pos__nth,axiom,
    ! [Na: nat,A3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ( aa(nat,real,power_power(real,R3),Na) = A3 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1365_realpow__pos__nth__unique,axiom,
    ! [Na: nat,A3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
       => ? [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X3)
            & ( aa(nat,real,power_power(real,X3),Na) = A3 )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2)
                  & ( aa(nat,real,power_power(real,Y2),Na) = A3 ) )
               => ( Y2 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1366_lemma__interval__lt,axiom,
    ! [A3: real,Xa: real,B3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B3)
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [Y2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xa),Y2))),D6)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),B3) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_1367_realpow__pos__nth2,axiom,
    ! [A3: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ? [R3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
          & ( aa(nat,real,power_power(real,R3),aa(nat,nat,suc,Na)) = A3 ) ) ) ).

% realpow_pos_nth2
tff(fact_1368_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(list(A),$o,P,Ys) )
         => aa(list(A),$o,P,Xs2) )
     => aa(list(A),$o,P,Xs) ) ).

% length_induct
tff(fact_1369_finite__maxlen,axiom,
    ! [A: $tType,M5: set(list(A))] :
      ( aa(set(list(A)),$o,finite_finite2(list(A)),M5)
     => ? [N2: nat] :
        ! [X2: list(A)] :
          ( member(list(A),X2,M5)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),X2)),N2) ) ) ).

% finite_maxlen
tff(fact_1370_sin__bound__lemma,axiom,
    ! [Xa: real,Ya: real,U: real,V2: real] :
      ( ( Xa = Ya )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),U)),Ya))),V2) ) ) ).

% sin_bound_lemma
tff(fact_1371_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A3: nat,B3: nat] :
      ( ! [A5: nat,B5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A5),B5)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B5),A5) )
     => ( ! [A5: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A5),zero_zero(nat))
       => ( ! [A5: nat,B5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A5),B5)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),B5)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A3),B3) ) ) ) ).

% Euclid_induct
tff(fact_1372_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B3: A,A3: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_1373_ln__root,axiom,
    ! [Na: nat,B3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Na),B3)) = divide_divide(real,aa(real,real,ln_ln(real),B3),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% ln_root
tff(fact_1374_log__of__power__le,axiom,
    ! [M: nat,B3: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,B3),Na))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_le
tff(fact_1375_decr__lemma,axiom,
    ! [D2: int,Xa: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,Xa),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,minus_minus(int,Xa),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_1376_incr__lemma,axiom,
    ! [D2: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),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,minus_minus(int,Xa),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_1377_linear__plus__1__le__power,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)),one_one(real))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),one_one(real))),Na)) ) ).

% linear_plus_1_le_power
tff(fact_1378_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_1379_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_1380_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_1381_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% mult_cancel_left
tff(fact_1382_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A3: A,C3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% mult_cancel_right
tff(fact_1383_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% mult_1
tff(fact_1384_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.right_neutral
tff(fact_1385_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Na: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) = 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),Na)) ) ).

% of_nat_mult
tff(fact_1386_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B3: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B3 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_1387_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_1388_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C3: A,B3: A] :
          ( ( C3 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3) )
        <=> ( ( C3 = zero_zero(A) )
            | ( B3 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_1389_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A,C3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = C3 )
        <=> ( ( C3 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_1390_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Ya)) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Ya = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_1391_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B3: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) = $ite(C3 = zero_zero(A),zero_zero(A),divide_divide(A,A3,B3)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1392_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1393_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1394_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1395_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1396_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),A3) = B3 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_1397_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),B3) = A3 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_1398_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) = divide_divide(A,A3,B3) ) ) ) ).

% div_mult_mult1
tff(fact_1399_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) = divide_divide(A,A3,B3) ) ) ) ).

% div_mult_mult2
tff(fact_1400_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C3: A,A3: A,B3: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) = $ite(C3 = zero_zero(A),zero_zero(A),divide_divide(A,A3,B3)) ) ).

% div_mult_mult1_if
tff(fact_1401_not__real__square__gt__zero,axiom,
    ! [Xa: real] :
      ( ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa))
    <=> ( Xa = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1402_real__root__Suc__0,axiom,
    ! [Xa: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),Xa) = Xa ).

% real_root_Suc_0
tff(fact_1403_root__0,axiom,
    ! [Xa: real] : aa(real,real,root(zero_zero(nat)),Xa) = zero_zero(real) ).

% root_0
tff(fact_1404_real__root__eq__iff,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = aa(real,real,root(Na),Ya) )
      <=> ( Xa = Ya ) ) ) ).

% real_root_eq_iff
tff(fact_1405_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = divide_divide(A,one_one(A),B3) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1406_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,B3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = divide_divide(A,one_one(A),A3) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1407_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A,C3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B3)) ) ) ) ).

% div_mult_self1
tff(fact_1408_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A,C3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B3)) ) ) ) ).

% div_mult_self2
tff(fact_1409_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B3)) ) ) ) ).

% div_mult_self3
tff(fact_1410_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)),A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),divide_divide(A,A3,B3)) ) ) ) ).

% div_mult_self4
tff(fact_1411_real__root__eq__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = zero_zero(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_1412_real__root__less__iff,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Ya))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ) ).

% real_root_less_iff
tff(fact_1413_real__root__le__iff,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Ya))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ) ).

% real_root_le_iff
tff(fact_1414_real__root__one,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_1415_real__root__eq__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(real,real,root(Na),Xa) = one_one(real) )
      <=> ( Xa = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_1416_log__eq__one,axiom,
    ! [A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),A3) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_1417_log__less__cancel__iff,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A3),Xa)),aa(real,real,log(A3),Ya))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_1418_log__less__one__cancel__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A3),Xa)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),A3) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_1419_one__less__log__cancel__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,log(A3),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xa) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_1420_log__less__zero__cancel__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(A3),Xa)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_1421_zero__less__log__cancel__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,log(A3),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_1422_real__root__gt__0__iff,axiom,
    ! [Na: nat,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Na),Ya))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya) ) ) ).

% real_root_gt_0_iff
tff(fact_1423_real__root__lt__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_1424_real__root__le__0__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_1425_real__root__ge__0__iff,axiom,
    ! [Na: nat,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Ya))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya) ) ) ).

% real_root_ge_0_iff
tff(fact_1426_real__root__gt__1__iff,axiom,
    ! [Na: nat,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,root(Na),Ya))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ya) ) ) ).

% real_root_gt_1_iff
tff(fact_1427_real__root__lt__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_1428_real__root__le__1__iff,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_1429_real__root__ge__1__iff,axiom,
    ! [Na: nat,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,root(Na),Ya))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Ya) ) ) ).

% real_root_ge_1_iff
tff(fact_1430_zero__le__log__cancel__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,log(A3),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_1431_log__le__zero__cancel__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A3),Xa)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_1432_one__le__log__cancel__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,log(A3),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Xa) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_1433_log__le__one__cancel__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A3),Xa)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),A3) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_1434_log__le__cancel__iff,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(A3),Xa)),aa(real,real,log(A3),Ya))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_1435_real__root__pow__pos2,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos2
tff(fact_1436_log__pow__cancel,axiom,
    ! [A3: real,B3: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),aa(nat,real,power_power(real,A3),B3)) = aa(nat,real,semiring_1_of_nat(real),B3) ) ) ) ).

% log_pow_cancel
tff(fact_1437_mult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B3: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) ) ).

% mult.left_commute
tff(fact_1438_mult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3) ) ).

% mult.commute
tff(fact_1439_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) ) ).

% mult.assoc
tff(fact_1440_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) ) ).

% ab_semigroup_mult_class.mult_ac(1)
tff(fact_1441_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) != zero_zero(A) )
         => ( ( A3 != zero_zero(A) )
            & ( B3 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_1442_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = zero_zero(A) )
         => ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_1443_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_1444_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3) )
          <=> ( A3 = B3 ) ) ) ) ).

% mult_left_cancel
tff(fact_1445_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3) )
          <=> ( A3 = B3 ) ) ) ) ).

% mult_right_cancel
tff(fact_1446_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.comm_neutral
tff(fact_1447_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_1448_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xa: nat,Ya: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xa)),Ya) = aa(A,A,aa(A,fun(A,A),times_times(A),Ya),aa(nat,A,semiring_1_of_nat(A),Xa)) ) ).

% mult_of_nat_commute
tff(fact_1449_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A3: A,B3: A,C3: A,D2: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A3 = B3 )
              & ( C3 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C3)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1450_log__base__root,axiom,
    ! [Na: nat,B3: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
       => ( aa(real,real,log(aa(real,real,root(Na),B3)),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B3),Xa)) ) ) ) ).

% log_base_root
tff(fact_1451_log__mult,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
           => ( aa(real,real,log(A3),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Ya)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A3),Xa)),aa(real,real,log(A3),Ya)) ) ) ) ) ) ).

% log_mult
tff(fact_1452_log__nat__power,axiom,
    ! [Xa: real,B3: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(B3),aa(nat,real,power_power(real,Xa),Na)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B3),Xa)) ) ) ).

% log_nat_power
tff(fact_1453_real__root__pos__pos__le,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ).

% real_root_pos_pos_le
tff(fact_1454_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1455_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1456_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1457_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_1458_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_1459_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_1460_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_1461_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1462_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) ) ) ) ).

% mult_right_mono
tff(fact_1463_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) ) ) ) ).

% mult_right_mono_neg
tff(fact_1464_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) ) ) ) ).

% mult_left_mono
tff(fact_1465_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_1466_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1467_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ).

% split_mult_pos_le
tff(fact_1468_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)) ) ).

% zero_le_square
tff(fact_1469_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1470_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_1471_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_neg_neg
tff(fact_1472_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),zero_zero(A)) ) ).

% not_square_less_zero
tff(fact_1473_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1474_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_1475_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_1476_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_pos_pos
tff(fact_1477_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_1478_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1479_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ) ).

% zero_less_mult_pos
tff(fact_1480_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1481_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1482_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1483_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1484_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) ) ) ) ).

% mult_strict_left_mono
tff(fact_1485_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1486_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1487_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) ) ) ) ).

% mult_strict_right_mono
tff(fact_1488_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1489_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1490_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,Na: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),Na)) ) ) ) ).

% less_1_mult
tff(fact_1491_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Ya: A,Z: A,Xa: A,W2: A] :
          ( ( Ya != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( divide_divide(A,Xa,Ya) = divide_divide(A,W2,Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W2),Ya) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1492_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C3: A,A3: A] :
          ( ( divide_divide(A,B3,C3) = A3 )
        <=> $ite(C3 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_1493_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( A3 = divide_divide(A,B3,C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_1494_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B3: A,A3: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) )
           => ( divide_divide(A,B3,C3) = A3 ) ) ) ) ).

% divide_eq_imp
tff(fact_1495_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B3 )
           => ( A3 = divide_divide(A,B3,C3) ) ) ) ) ).

% eq_divide_imp
tff(fact_1496_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,B3: A,A3: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( divide_divide(A,B3,C3) = A3 )
          <=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1497_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: A,A3: A,B3: A] :
          ( ( C3 != zero_zero(A) )
         => ( ( A3 = divide_divide(A,B3,C3) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = B3 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1498_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A,B3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),B3)),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D2)) ) ) ) ).

% abs_mult_less
tff(fact_1499_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1500_log__eq__div__ln__mult__log,axiom,
    ! [A3: real,B3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
         => ( ( B3 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
             => ( aa(real,real,log(A3),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),B3),aa(real,real,ln_ln(real),A3))),aa(real,real,log(B3),Xa)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_1501_log__root,axiom,
    ! [Na: nat,A3: real,B3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
       => ( aa(real,real,log(B3),aa(real,real,root(Na),A3)) = divide_divide(real,aa(real,real,log(B3),A3),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_root
tff(fact_1502_real__root__less__mono,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Ya)) ) ) ).

% real_root_less_mono
tff(fact_1503_real__root__le__mono,axiom,
    ! [Na: nat,Xa: real,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(Na),Ya)) ) ) ).

% real_root_le_mono
tff(fact_1504_real__root__power,axiom,
    ! [Na: nat,Xa: real,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(nat,real,power_power(real,Xa),K)) = aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),K) ) ) ).

% real_root_power
tff(fact_1505_real__root__abs,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,abs_abs(real),Xa)) = aa(real,real,abs_abs(real),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_abs
tff(fact_1506_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1507_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1508_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1509_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1510_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1511_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1512_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1513_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1514_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1515_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1516_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1517_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1518_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1519_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1520_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),A3) ) ) ) ).

% mult_left_le
tff(fact_1521_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_1522_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Ya)),Xa) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1523_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Xa)),Xa) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1524_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Ya))),zero_zero(A))
        <=> ( ( Xa = zero_zero(A) )
            & ( Ya = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1525_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Ya))) ) ).

% sum_squares_ge_zero
tff(fact_1526_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Ya)))
        <=> ( ( Xa != zero_zero(A) )
            | ( Ya != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1527_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xa: A,Ya: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Ya))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1528_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) = divide_divide(A,divide_divide(A,A3,B3),C3) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1529_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C3)),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% divide_less_eq
tff(fact_1530_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1531_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C3)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3) ) ) ) ).

% neg_divide_less_eq
tff(fact_1532_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1533_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C3)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1534_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3) ) ) ) ).

% pos_less_divide_eq
tff(fact_1535_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ya: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Ya))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Ya)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1536_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ya: A,Z: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Ya)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),divide_divide(A,Xa,Ya)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1537_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B3)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1538_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B3)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1539_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C3: A,B3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),E2)),C3)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1540_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C3: A,B3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),A3)),E2)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1541_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,Z)),B3) = $ite(Z = zero_zero(A),B3,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z)),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1542_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),divide_divide(A,B3,Z)) = $ite(Z = zero_zero(A),A3,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z)),B3),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1543_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Ya: A,Z: A,Xa: A,W2: A] :
          ( ( Ya != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xa,Ya)),divide_divide(A,W2,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Ya)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1544_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Ya: A,Xa: A,Z: A] :
          ( ( Ya != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xa,Ya)),Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Ya)),Ya) ) ) ) ).

% add_frac_num
tff(fact_1545_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Ya: A,Z: A,Xa: A] :
          ( ( Ya != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,Xa,Ya)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Ya)),Ya) ) ) ) ).

% add_num_frac
tff(fact_1546_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),divide_divide(A,Ya,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),Ya),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1547_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Xa,Z)),Ya) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z)),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1548_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C3: A,B3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),A3)),E2)),D2)) ) ) ).

% less_add_iff2
tff(fact_1549_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E2: A,C3: A,B3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E2)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),E2)),C3)),D2) ) ) ).

% less_add_iff1
tff(fact_1550_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,Z: A] :
          aa(A,A,minus_minus(A,A3),divide_divide(A,B3,Z)) = $ite(Z = zero_zero(A),A3,divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z)),B3),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1551_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Ya: A,Z: A,Xa: A,W2: A] :
          ( ( Ya != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,divide_divide(A,Xa,Ya)),divide_divide(A,W2,Z)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Ya)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1552_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,Xa),divide_divide(A,Ya,Z)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),Ya),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1553_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,divide_divide(A,Xa,Z)),Ya) = divide_divide(A,aa(A,A,minus_minus(A,Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z)),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1554_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),Xa)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1555_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),Na))) ) ) ).

% power_less_power_Suc
tff(fact_1556_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),Na))) ) ) ).

% power_gt1_lemma
tff(fact_1557_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3)) ) ) ) ).

% abs_eq_mult
tff(fact_1558_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Ya)),Xa) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Xa)) ) ) ) ).

% abs_mult_pos
tff(fact_1559_reals__Archimedean3,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ! [Y2: real] :
        ? [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),Xa)) ) ).

% reals_Archimedean3
tff(fact_1560_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Na) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( Na = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1561_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int,K2: int] :
            ( aa(int,$o,P1,X3)
          <=> aa(int,$o,P1,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Z3)
             => ( aa(int,$o,P,X3)
              <=> aa(int,$o,P1,X3) ) )
         => ( ? [X_12: int] : aa(int,$o,P1,X_12)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_1562_plusinfinity,axiom,
    ! [D2: int,P2: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int,K2: int] :
            ( aa(int,$o,P2,X3)
          <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X3)
             => ( aa(int,$o,P,X3)
              <=> aa(int,$o,P2,X3) ) )
         => ( ? [X_12: int] : aa(int,$o,P2,X_12)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_1563_zdiv__zmult2__eq,axiom,
    ! [C3: int,A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C3)
     => ( divide_divide(int,A3,aa(int,int,aa(int,fun(int,int),times_times(int),B3),C3)) = divide_divide(int,divide_divide(int,A3,B3),C3) ) ) ).

% zdiv_zmult2_eq
tff(fact_1564_real__root__gt__zero,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_gt_zero
tff(fact_1565_real__root__strict__decreasing,axiom,
    ! [Na: nat,N4: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(N4),Xa)),aa(real,real,root(Na),Xa)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_1566_log__base__change,axiom,
    ! [A3: real,B3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(B3),Xa) = divide_divide(real,aa(real,real,log(A3),Xa),aa(real,real,log(A3),B3)) ) ) ) ).

% log_base_change
tff(fact_1567_root__abs__power,axiom,
    ! [Na: nat,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Na),aa(nat,real,power_power(real,Ya),Na))) = aa(real,real,abs_abs(real),Ya) ) ) ).

% root_abs_power
tff(fact_1568_less__log__of__power,axiom,
    ! [B3: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,B3),Na)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B3),M)) ) ) ).

% less_log_of_power
tff(fact_1569_log__of__power__eq,axiom,
    ! [M: nat,B3: real,Na: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,power_power(real,B3),Na) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
       => ( aa(nat,real,semiring_1_of_nat(real),Na) = aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_1570_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( ! [Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Xa)),Ya) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ).

% field_le_mult_one_interval
tff(fact_1571_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1572_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1573_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1574_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1575_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1576_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1577_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),C3)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1578_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1579_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B3)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1580_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ya: A,Z: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Ya)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),divide_divide(A,Xa,Ya)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1581_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ya: A,Xa: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Ya))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Ya)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1582_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3) ) ) ) ).

% pos_le_divide_eq
tff(fact_1583_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C3)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1584_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,C3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1585_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C3)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3) ) ) ) ).

% neg_divide_le_eq
tff(fact_1586_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,C3,A3)),divide_divide(A,C3,B3)) ) ) ) ) ).

% divide_left_mono
tff(fact_1587_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1588_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C3)),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% divide_le_eq
tff(fact_1589_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xa: A,A3: A,Ya: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Ya))),A3) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1590_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ya: A,Z: A,Xa: A,W2: A] :
          ( ( Ya != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,Xa,Ya)),divide_divide(A,W2,Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Ya)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_1591_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Ya: A,Z: A,Xa: A,W2: A] :
          ( ( Ya != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,Xa,Ya)),divide_divide(A,W2,Z))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W2),Ya)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_1592_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,A3),Na))),aa(nat,A,power_power(A,A3),Na)) ) ) ) ).

% power_Suc_less
tff(fact_1593_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1594_q__pos__lemma,axiom,
    ! [B9: int,Q4: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B9),Q4)),R4))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B9)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B9)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q4) ) ) ) ).

% q_pos_lemma
tff(fact_1595_zdiv__mono2__lemma,axiom,
    ! [B3: int,Q5: int,R2: int,B9: int,Q4: 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),B3),Q5)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B9),Q4)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B9),Q4)),R4))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B9)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B9)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B9),B3)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q4) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1596_zdiv__mono2__neg__lemma,axiom,
    ! [B3: int,Q5: int,R2: int,B9: int,Q4: 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),B3),Q5)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B9),Q4)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B9),Q4)),R4)),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B3)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B9)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B9),B3)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1597_unique__quotient__lemma,axiom,
    ! [B3: int,Q4: int,R4: int,Q5: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R4),B3)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q4),Q5) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1598_unique__quotient__lemma__neg,axiom,
    ! [B3: int,Q4: int,R4: int,Q5: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q4)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R2))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),R2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),R4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q4) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1599_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int] :
            ( aa(int,$o,P,X3)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X2: int] :
              ( aa(int,$o,P,X2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1600_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int] :
            ( aa(int,$o,P,X3)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X3),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
         => ! [X2: int] :
              ( aa(int,$o,P,X2)
             => aa(int,$o,P,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1601_ln__mult,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Ya)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),Xa)),aa(real,real,ln_ln(real),Ya)) ) ) ) ).

% ln_mult
tff(fact_1602_real__root__pos__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Na),Xa)) ) ) ).

% real_root_pos_pos
tff(fact_1603_real__root__strict__increasing,axiom,
    ! [Na: nat,N4: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),N4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,root(Na),Xa)),aa(real,real,root(N4),Xa)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_1604_real__root__decreasing,axiom,
    ! [Na: nat,N4: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N4),Xa)),aa(real,real,root(Na),Xa)) ) ) ) ).

% real_root_decreasing
tff(fact_1605_real__root__pow__pos,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),Na) = Xa ) ) ) ).

% real_root_pow_pos
tff(fact_1606_real__root__power__cancel,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,real,root(Na),aa(nat,real,power_power(real,Xa),Na)) = Xa ) ) ) ).

% real_root_power_cancel
tff(fact_1607_real__root__pos__unique,axiom,
    ! [Na: nat,Ya: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( ( aa(nat,real,power_power(real,Ya),Na) = Xa )
         => ( aa(real,real,root(Na),Xa) = Ya ) ) ) ) ).

% real_root_pos_unique
tff(fact_1608_log__divide,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
           => ( aa(real,real,log(A3),divide_divide(real,Xa,Ya)) = aa(real,real,minus_minus(real,aa(real,real,log(A3),Xa)),aa(real,real,log(A3),Ya)) ) ) ) ) ) ).

% log_divide
tff(fact_1609_le__log__of__power,axiom,
    ! [B3: real,Na: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,B3),Na)),M)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(B3),M)) ) ) ).

% le_log_of_power
tff(fact_1610_log__base__pow,axiom,
    ! [A3: real,Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( aa(real,real,log(aa(nat,real,power_power(real,A3),Na)),Xa) = divide_divide(real,aa(real,real,log(A3),Xa),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log_base_pow
tff(fact_1611_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xa: A,A3: A,Ya: A,U: A,V2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),V2)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V2) = one_one(A) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),V2),Ya))),A3) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1612_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V2: A,R2: A,S3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),S3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,minus_minus(A,V2),U)),S3))),V2) ) ) ) ) ).

% scaling_mono
tff(fact_1613_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P3: A,M: nat] :
          aa(nat,A,power_power(A,P3),M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(nat,A,power_power(A,P3),aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1614_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Na: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A3),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))),A3) = aa(nat,A,power_power(A,A3),Na) ) ) ) ).

% power_minus_mult
tff(fact_1615_real__archimedian__rdiv__eq__0,axiom,
    ! [Xa: real,C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C3)
       => ( ! [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M4)),Xa)),C3) )
         => ( Xa = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1616_split__zdiv,axiom,
    ! [P: fun(int,$o),Na: int,K: int] :
      ( aa(int,$o,P,divide_divide(int,Na,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_1617_int__div__neg__eq,axiom,
    ! [A3: int,B3: int,Q5: int,R2: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),R2)
         => ( divide_divide(int,A3,B3) = Q5 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1618_int__div__pos__eq,axiom,
    ! [A3: int,B3: int,Q5: int,R2: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B3)
         => ( divide_divide(int,A3,B3) = Q5 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1619_ln__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(nat,real,power_power(real,Xa),Na)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,ln_ln(real),Xa)) ) ) ).

% ln_realpow
tff(fact_1620_real__root__increasing,axiom,
    ! [Na: nat,N4: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),N4)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Na),Xa)),aa(real,real,root(N4),Xa)) ) ) ) ) ).

% real_root_increasing
tff(fact_1621_log__of__power__less,axiom,
    ! [M: nat,B3: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,B3),Na))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ) ).

% log_of_power_less
tff(fact_1622_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Ya))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1623_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1624_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Z)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya) ) ) ) ).

% mult_less_iff1
tff(fact_1625_arctan__add,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Ya)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),arctan(Xa)),arctan(Ya)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Ya),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Ya)))) ) ) ) ).

% arctan_add
tff(fact_1626_root__powr__inverse,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,real,root(Na),Xa) = powr(real,Xa,divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Na))) ) ) ) ).

% root_powr_inverse
tff(fact_1627_split__root,axiom,
    ! [P: fun(real,$o),Na: nat,Xa: real] :
      ( aa(real,$o,P,aa(real,real,root(Na),Xa))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ! [Y4: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y4)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Y4)),Na)) = Xa )
             => aa(real,$o,P,Y4) ) ) ) ) ).

% split_root
tff(fact_1628_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( gbinomial(A,A3,K) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),K))),gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A)),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_1629_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( divide_divide(int,K,L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_1630_uminus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( uminus(A)
     => ! [A4: fun(B,A),Xa: B] : aa(B,A,aa(fun(B,A),fun(B,A),uminus_uminus(fun(B,A)),A4),Xa) = aa(A,A,uminus_uminus(A),aa(B,A,A4,Xa)) ) ).

% uminus_apply
tff(fact_1631_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),Xa)) = Xa ) ).

% boolean_algebra_class.boolean_algebra.double_compl
tff(fact_1632_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,uminus_uminus(A),Xa) = aa(A,A,uminus_uminus(A),Ya) )
        <=> ( Xa = Ya ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
tff(fact_1633_add_Oinverse__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),aa(A,A,uminus_uminus(A),A3)) = A3 ) ).

% add.inverse_inverse
tff(fact_1634_neg__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,uminus_uminus(A),B3) )
        <=> ( A3 = B3 ) ) ) ).

% neg_equal_iff_equal
tff(fact_1635_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% neg_le_iff_le
tff(fact_1636_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Ya))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ).

% compl_le_compl_iff
tff(fact_1637_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = A3 )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_1638_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_1639_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_1640_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A3) )
        <=> ( zero_zero(A) = A3 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_1641_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_1642_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Ya))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa) ) ) ).

% compl_less_compl_iff
tff(fact_1643_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% neg_less_iff_less
tff(fact_1644_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B3)) = B3 ) ).

% add_minus_cancel
tff(fact_1645_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = B3 ) ).

% minus_add_cancel
tff(fact_1646_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) ) ).

% minus_add_distrib
tff(fact_1647_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A3),B3)) = aa(A,A,minus_minus(A,B3),A3) ) ).

% minus_diff_eq
tff(fact_1648_abs__minus__cancel,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,abs_abs(A),A3) ) ).

% abs_minus_cancel
tff(fact_1649_mult__cancel2,axiom,
    ! [M: nat,K: nat,Na: 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),Na),K) )
    <=> ( ( M = Na )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_1650_mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: 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),Na) )
    <=> ( ( M = Na )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_1651_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_1652_mult__is__0,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( Na = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_1653_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_1654_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W2: A,Z: A] :
          ( ( powr(A,W2,Z) = zero_zero(A) )
        <=> ( W2 = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_1655_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_1656_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
    <=> ( ( M = one_one(nat) )
        & ( Na = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_1657_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( Na = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_1658_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_1659_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% neg_le_0_iff_le
tff(fact_1660_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_1661_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% neg_less_eq_nonneg
tff(fact_1662_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_1663_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% neg_less_pos
tff(fact_1664_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_1665_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% neg_less_0_iff_less
tff(fact_1666_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_1667_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_1668_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,zero_zero(A)),A3) = aa(A,A,uminus_uminus(A),A3) ) ).

% diff_0
tff(fact_1669_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B3: A] : aa(A,A,minus_minus(A,zero_zero(A)),B3) = aa(A,A,uminus_uminus(A),B3) ) ).

% verit_minus_simplify(3)
tff(fact_1670_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_1671_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_1672_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) ) ).

% diff_minus_eq_add
tff(fact_1673_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,minus_minus(A,B3),A3) ) ).

% uminus_add_conv_diff
tff(fact_1674_inf__compl__bot__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)) = bot_bot(A) ) ).

% inf_compl_bot_left1
tff(fact_1675_inf__compl__bot__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),Ya)) = bot_bot(A) ) ).

% inf_compl_bot_left2
tff(fact_1676_inf__compl__bot__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,aa(A,fun(A,A),inf_inf(A),Ya),aa(A,A,uminus_uminus(A),Xa))) = bot_bot(A) ) ).

% inf_compl_bot_right
tff(fact_1677_boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),Xa) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_left
tff(fact_1678_boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Xa)) = bot_bot(A) ) ).

% boolean_algebra.conj_cancel_right
tff(fact_1679_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_1680_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),sgn_sgn(A,A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% sgn_greater
tff(fact_1681_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),sgn_sgn(A,A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_1682_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Ya)) ) ).

% boolean_algebra.de_Morgan_conj
tff(fact_1683_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Ya)) ) ).

% boolean_algebra.de_Morgan_disj
tff(fact_1684_one__eq__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_1685_mult__eq__1__iff,axiom,
    ! [M: nat,Na: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_1686_nat__0__less__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ).

% nat_0_less_mult_iff
tff(fact_1687_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% mult_less_cancel2
tff(fact_1688_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A] :
          powr(A,Xa,zero_zero(A)) = $ite(Xa = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_1689_mult__Suc__right,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc_right
tff(fact_1690_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K: nat] : gbinomial(A,zero_zero(A),aa(nat,nat,suc,K)) = zero_zero(A) ) ).

% gbinomial_0(2)
tff(fact_1691_negative__eq__positive,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( Na = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_1692_powr__gt__zero,axiom,
    ! [Xa: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),powr(real,Xa,A3))
    <=> ( Xa != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_1693_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : gbinomial(A,A3,zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_1694_powr__nonneg__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,A3,Xa)),zero_zero(real))
    <=> ( A3 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_1695_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : gbinomial(A,A3,aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% gbinomial_Suc0
tff(fact_1696_powr__less__cancel__iff,axiom,
    ! [Xa: real,A3: real,B3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A3)),powr(real,Xa,B3))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3) ) ) ).

% powr_less_cancel_iff
tff(fact_1697_negative__zle,axiom,
    ! [Na: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zle
tff(fact_1698_arctan__less__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_1699_zero__less__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),arctan(Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% zero_less_arctan_iff
tff(fact_1700_zero__le__arctan__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arctan(Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% zero_le_arctan_iff
tff(fact_1701_arctan__le__zero__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_1702_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_1703_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_1704_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_1705_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(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_1706_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_nonpos
tff(fact_1707_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( sgn_sgn(A,A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_1708_one__le__mult__iff,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Na) ) ) ).

% one_le_mult_iff
tff(fact_1709_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_1710_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na),K))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% mult_le_cancel2
tff(fact_1711_div__mult__self1__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M),Na) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1712_div__mult__self__is__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na),Na) = M ) ) ).

% div_mult_self_is_m
tff(fact_1713_powr__eq__one__iff,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
     => ( ( powr(real,A3,Xa) = one_one(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_1714_powr__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,one_one(real)) = Xa ) ) ).

% powr_one
tff(fact_1715_powr__one__gt__zero__iff,axiom,
    ! [Xa: real] :
      ( ( powr(real,Xa,one_one(real)) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% powr_one_gt_zero_iff
tff(fact_1716_negative__zless,axiom,
    ! [Na: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zless
tff(fact_1717_powr__le__cancel__iff,axiom,
    ! [Xa: real,A3: real,B3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A3)),powr(real,Xa,B3))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3) ) ) ).

% powr_le_cancel_iff
tff(fact_1718_nat__zminus__int,axiom,
    ! [Na: nat] : aa(int,nat,nat2,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_1719_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( sgn_sgn(A,A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_1720_powr__log__cancel,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( powr(real,A3,aa(real,real,log(A3),Xa)) = Xa ) ) ) ) ).

% powr_log_cancel
tff(fact_1721_log__powr__cancel,axiom,
    ! [A3: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),powr(real,A3,Ya)) = Ya ) ) ) ).

% log_powr_cancel
tff(fact_1722_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( sgn_sgn(A,B3) != sgn_sgn(A,A3) )
         => ( ( sgn_sgn(A,A3) != zero_zero(A) )
           => ( ( sgn_sgn(A,B3) != zero_zero(A) )
             => ( sgn_sgn(A,A3) = aa(A,A,uminus_uminus(A),sgn_sgn(A,B3)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_1723_fun__Compl__def,axiom,
    ! [B: $tType,A: $tType] :
      ( uminus(B)
     => ! [A4: fun(A,B),X2: A] : aa(A,B,aa(fun(A,B),fun(A,B),uminus_uminus(fun(A,B)),A4),X2) = aa(B,B,uminus_uminus(B),aa(A,B,A4,X2)) ) ).

% fun_Compl_def
tff(fact_1724_equation__minus__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B3) )
        <=> ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% equation_minus_iff
tff(fact_1725_minus__equation__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B3 )
        <=> ( aa(A,A,uminus_uminus(A),B3) = A3 ) ) ) ).

% minus_equation_iff
tff(fact_1726_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( sgn_sgn(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_1727_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( sgn_sgn(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_1728_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_imp_neg_le
tff(fact_1729_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),A3) ) ) ).

% minus_le_iff
tff(fact_1730_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_minus_iff
tff(fact_1731_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ya)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),Ya) ) ) ).

% compl_le_swap2
tff(fact_1732_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),aa(A,A,uminus_uminus(A),Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),Ya)) ) ) ).

% compl_le_swap1
tff(fact_1733_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Ya)),aa(A,A,uminus_uminus(A),Xa)) ) ) ).

% compl_mono
tff(fact_1734_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),aa(A,A,uminus_uminus(A),Xa))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,uminus_uminus(A),Ya)) ) ) ).

% compl_less_swap1
tff(fact_1735_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Ya)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),Xa)),Ya) ) ) ).

% compl_less_swap2
tff(fact_1736_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% less_minus_iff
tff(fact_1737_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),A3) ) ) ).

% minus_less_iff
tff(fact_1738_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_1739_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_1740_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ).

% is_num_normalize(8)
tff(fact_1741_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A4: A,K: A,A3: A] :
          ( ( A4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A3) )
         => ( aa(A,A,uminus_uminus(A),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A3)) ) ) ) ).

% group_cancel.neg1
tff(fact_1742_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ).

% add.inverse_distrib_swap
tff(fact_1743_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B3: A,A3: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),B3)),A3) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A3)),B3) ) ).

% minus_diff_commute
tff(fact_1744_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Na: 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)),Na) )
    <=> ( M = Na ) ) ).

% Suc_mult_cancel1
tff(fact_1745_mult__0,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),Na) = zero_zero(nat) ).

% mult_0
tff(fact_1746_arctan__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xa)),arctan(Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ).

% arctan_less_iff
tff(fact_1747_arctan__monotone,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Xa)),arctan(Ya)) ) ).

% arctan_monotone
tff(fact_1748_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_1749_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_1750_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_1751_le__square,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_1752_le__cube,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),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_1753_arctan__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),arctan(Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ).

% arctan_le_iff
tff(fact_1754_arctan__monotone_H,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xa)),arctan(Ya)) ) ).

% arctan_monotone'
tff(fact_1755_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = 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),Na)) ).

% add_mult_distrib2
tff(fact_1756_add__mult__distrib,axiom,
    ! [M: nat,Na: 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),Na)),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),Na),K)) ).

% add_mult_distrib
tff(fact_1757_diff__mult__distrib,axiom,
    ! [M: nat,Na: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),Na)),K) = aa(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),Na),K)) ).

% diff_mult_distrib
tff(fact_1758_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,minus_minus(nat,M),Na)) = aa(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),Na)) ).

% diff_mult_distrib2
tff(fact_1759_nat__mult__1,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),Na) = Na ).

% nat_mult_1
tff(fact_1760_nat__mult__1__right,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),one_one(nat)) = Na ).

% nat_mult_1_right
tff(fact_1761_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          sgn_sgn(A,Xa) = $ite(
            Xa = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_1762_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( sgn_sgn(A,A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_1763_powr__non__neg,axiom,
    ! [A3: real,Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,A3,Xa)),zero_zero(real)) ).

% powr_non_neg
tff(fact_1764_powr__less__mono2__neg,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Ya,A3)),powr(real,Xa,A3)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_1765_powr__ge__pzero,axiom,
    ! [Xa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),powr(real,Xa,Ya)) ).

% powr_ge_pzero
tff(fact_1766_powr__mono2,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A3)),powr(real,Ya,A3)) ) ) ) ).

% powr_mono2
tff(fact_1767_powr__less__cancel,axiom,
    ! [Xa: real,A3: real,B3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A3)),powr(real,Xa,B3))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3) ) ) ).

% powr_less_cancel
tff(fact_1768_powr__less__mono,axiom,
    ! [A3: real,B3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A3)),powr(real,Xa,B3)) ) ) ).

% powr_less_mono
tff(fact_1769_powr__mono,axiom,
    ! [A3: real,B3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A3)),powr(real,Xa,B3)) ) ) ).

% powr_mono
tff(fact_1770_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),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_1771_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_1772_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_1773_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_1774_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B3 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_1775_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_1776_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A3) = B3 ) ) ) ).

% add.inverse_unique
tff(fact_1777_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_1778_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) )
        <=> ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% add_eq_0_iff
tff(fact_1779_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_1780_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_1781_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( divide_divide(A,aa(A,A,uminus_uminus(A),A3),aa(A,A,uminus_uminus(A),B3)) = divide_divide(A,A3,B3) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_1782_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),divide_divide(A,A3,B3)) = divide_divide(A,A3,aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_1783_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na),K) = gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Na),aa(nat,nat,minus_minus(nat,Na),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_1784_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_1785_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ).

% diff_conv_add_uminus
tff(fact_1786_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,K: A,B3: A,A3: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B3) )
         => ( aa(A,A,minus_minus(A,A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,minus_minus(A,A3),B3)) ) ) ) ).

% group_cancel.sub2
tff(fact_1787_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B3) ) ) ) ).

% abs_leI
tff(fact_1788_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ) ).

% abs_le_D2
tff(fact_1789_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ) ) ).

% abs_le_iff
tff(fact_1790_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,abs_abs(A),A3)) ) ).

% abs_ge_minus_self
tff(fact_1791_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),A3)),B3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ) ) ).

% abs_less_iff
tff(fact_1792_diff__eq,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] : aa(A,A,minus_minus(A,Xa),Ya) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Ya)) ) ).

% diff_eq
tff(fact_1793_inf__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),B3)) = bot_bot(A) ) ).

% inf_cancel_left1
tff(fact_1794_inf__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,uminus_uminus(A),Xa)),A3)),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),B3)) = bot_bot(A) ) ).

% inf_cancel_left2
tff(fact_1795_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% Suc_mult_less_cancel1
tff(fact_1796_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ) ).

% mult_less_mono2
tff(fact_1797_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ) ).

% mult_less_mono1
tff(fact_1798_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% Suc_mult_le_cancel1
tff(fact_1799_mult__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ).

% mult_Suc
tff(fact_1800_mult__eq__self__implies__10,axiom,
    ! [M: nat,Na: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) )
     => ( ( Na = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1801_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,M,Na)),I) ) ).

% less_mult_imp_div_less
tff(fact_1802_not__int__zless__negative,axiom,
    ! [Na: nat,M: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))) ).

% not_int_zless_negative
tff(fact_1803_div__times__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,Na)),Na)),M) ).

% div_times_less_eq_dividend
tff(fact_1804_times__div__less__eq__dividend,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),divide_divide(nat,M,Na))),M) ).

% times_div_less_eq_dividend
tff(fact_1805_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,divide_divide(A,A3,aa(nat,A,semiring_1_of_nat(A),K))),K)),gbinomial(A,A3,K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_1806_powr__less__mono2,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,Xa,A3)),powr(real,Ya,A3)) ) ) ) ).

% powr_less_mono2
tff(fact_1807_powr__mono2_H,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Ya,A3)),powr(real,Xa,A3)) ) ) ) ).

% powr_mono2'
tff(fact_1808_gr__one__powr,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),powr(real,Xa,Ya)) ) ) ).

% gr_one_powr
tff(fact_1809_powr__inj,axiom,
    ! [A3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( ( powr(real,A3,Xa) = powr(real,A3,Ya) )
        <=> ( Xa = Ya ) ) ) ) ).

% powr_inj
tff(fact_1810_powr__le1,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A3)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_1811_powr__mono__both,axiom,
    ! [A3: real,B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xa,A3)),powr(real,Ya,B3)) ) ) ) ) ).

% powr_mono_both
tff(fact_1812_ge__one__powr__ge__zero,axiom,
    ! [Xa: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),powr(real,Xa,A3)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_1813_powr__divide,axiom,
    ! [Xa: real,Ya: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( powr(real,divide_divide(real,Xa,Ya),A3) = divide_divide(real,powr(real,Xa,A3),powr(real,Ya,A3)) ) ) ) ).

% powr_divide
tff(fact_1814_powr__mult,axiom,
    ! [Xa: real,Ya: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Ya),A3) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Xa,A3)),powr(real,Ya,A3)) ) ) ) ).

% powr_mult
tff(fact_1815_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( sgn_sgn(A,A3) = one_one(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% sgn_1_pos
tff(fact_1816_sgn__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( sgn_sgn(real,aa(real,real,root(Na),Xa)) = sgn_sgn(real,Xa) ) ) ).

% sgn_root
tff(fact_1817_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_1818_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_1819_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_1820_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_1821_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_1822_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( C3 = aa(A,A,uminus_uminus(A),divide_divide(A,A3,B3)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_1823_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A,C3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),divide_divide(A,A3,B3)) = C3 )
          <=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_1824_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C3: A,A3: A] :
          ( ( aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3)) = A3 )
        <=> $ite(C3 != zero_zero(A),aa(A,A,uminus_uminus(A),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3),A3 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_1825_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3)) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) = aa(A,A,uminus_uminus(A),B3),A3 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_1826_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( divide_divide(A,A3,B3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_1827_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A3))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_1828_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,abs_abs(A),A3) = B3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
            & ( ( A3 = B3 )
              | ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_1829_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,abs_abs(A),B3) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            & ( ( B3 = A3 )
              | ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_1830_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_neg
tff(fact_1831_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X2: A] :
          aa(A,A,abs_abs(A),X2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),zero_zero(A)),aa(A,A,uminus_uminus(A),X2),X2) ) ).

% abs_if_raw
tff(fact_1832_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          aa(A,A,abs_abs(A),A3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)),aa(A,A,uminus_uminus(A),A3),A3) ) ).

% abs_if
tff(fact_1833_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) = bot_bot(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),Ya)) ) ) ).

% inf_shunt
tff(fact_1834_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Ya)),Z)) ) ) ).

% shunt1
tff(fact_1835_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(A,A,uminus_uminus(A),Ya))),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),Ya),Z)) ) ) ).

% shunt2
tff(fact_1836_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P3: A,Q5: A,R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q5),R2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P3),aa(A,A,uminus_uminus(A),Q5))),R2) ) ) ).

% sup_neg_inf
tff(fact_1837_n__less__n__mult__m,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)) ) ) ).

% n_less_n_mult_m
tff(fact_1838_n__less__m__mult__n,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ) ) ).

% n_less_m_mult_n
tff(fact_1839_one__less__mult,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)) ) ) ).

% one_less_mult
tff(fact_1840_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),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_1841_int__zle__neg,axiom,
    ! [Na: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))
    <=> ( ( Na = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_1842_div__less__iff__less__mult,axiom,
    ! [Q5: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,M,Q5)),Na)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)) ) ) ).

% div_less_iff_less_mult
tff(fact_1843_negative__zle__0,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Na))),zero_zero(int)) ).

% negative_zle_0
tff(fact_1844_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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_1845_zabs__def,axiom,
    ! [I: int] :
      aa(int,int,abs_abs(int),I) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_1846_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A3,M)),gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A3,K)),gbinomial(A,aa(A,A,minus_minus(A,A3),aa(nat,A,semiring_1_of_nat(A),K)),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_1847_powr__realpow,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(nat,real,semiring_1_of_nat(real),Na)) = aa(nat,real,power_power(real,Xa),Na) ) ) ).

% powr_realpow
tff(fact_1848_powr__less__iff,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B3,Ya)),Xa)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),aa(real,real,log(B3),Xa)) ) ) ) ).

% powr_less_iff
tff(fact_1849_less__powr__iff,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B3,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B3),Xa)),Ya) ) ) ) ).

% less_powr_iff
tff(fact_1850_log__less__iff,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(B3),Xa)),Ya)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B3,Ya)) ) ) ) ).

% log_less_iff
tff(fact_1851_less__log__iff,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),aa(real,real,log(B3),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B3,Ya)),Xa) ) ) ) ).

% less_log_iff
tff(fact_1852_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_1853_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3))),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)) ) ) ) ).

% minus_divide_less_eq
tff(fact_1854_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_1855_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_1856_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_1857_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_1858_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,Xa,Z))),Ya) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z)),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_1859_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z))),B3) = $ite(Z = zero_zero(A),B3,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z)),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_1860_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,Xa,Z))),Ya) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),Z)),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_1861_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,minus_minus(A,divide_divide(A,A3,Z)),B3) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B3),divide_divide(A,aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z)),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_1862_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,A3,Z))),B3) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B3),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z)),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_1863_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),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)) )
             => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ) ).

% int_cases3
tff(fact_1864_not__zle__0__negative,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))) ).

% not_zle_0_negative
tff(fact_1865_negative__zless__0,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Na)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_1866_negD,axiom,
    ! [Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),zero_zero(int))
     => ? [N2: nat] : Xa = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_1867_div__nat__eqI,axiom,
    ! [Na: nat,Q5: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q5)))
       => ( divide_divide(nat,M,Na) = Q5 ) ) ) ).

% div_nat_eqI
tff(fact_1868_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q5: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Q5)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),divide_divide(nat,Na,Q5))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)),Na) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1869_split__div,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,divide_divide(nat,M,Na))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Na != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I4)),J3) )
               => aa(nat,$o,P,I4) ) ) ) ) ) ).

% split_div
tff(fact_1870_dividend__less__div__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,Na)),Na))) ) ).

% dividend_less_div_times
tff(fact_1871_dividend__less__times__div,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),divide_divide(nat,M,Na)))) ) ).

% dividend_less_times_div
tff(fact_1872_mult__eq__if,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),Na))) ).

% mult_eq_if
tff(fact_1873_verit__less__mono__div__int2,axiom,
    ! [A4: int,B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A4),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),Na))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),divide_divide(int,B2,Na)),divide_divide(int,A4,Na)) ) ) ).

% verit_less_mono_div_int2
tff(fact_1874_div__eq__minus1,axiom,
    ! [B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),B3) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_1875_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,nat,nat2,aa(int,int,aa(int,fun(int,int),times_times(int),Z),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_1876_powr__mult__base,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,Xa,Ya)) = powr(real,Xa,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Ya)) ) ) ).

% powr_mult_base
tff(fact_1877_sgn__power__injE,axiom,
    ! [A3: real,Na: nat,Xa: real,B3: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,A3)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),A3)),Na)) = Xa )
     => ( ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,B3)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),B3)),Na)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( A3 = B3 ) ) ) ) ).

% sgn_power_injE
tff(fact_1878_le__log__iff,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),aa(real,real,log(B3),Xa))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B3,Ya)),Xa) ) ) ) ).

% le_log_iff
tff(fact_1879_log__le__iff,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B3),Xa)),Ya)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B3,Ya)) ) ) ) ).

% log_le_iff
tff(fact_1880_le__powr__iff,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B3,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(B3),Xa)),Ya) ) ) ) ).

% le_powr_iff
tff(fact_1881_powr__le__iff,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B3,Ya)),Xa)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),aa(real,real,log(B3),Xa)) ) ) ) ).

% powr_le_iff
tff(fact_1882_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_1883_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3))),A3)
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)) ) ) ) ).

% minus_divide_le_eq
tff(fact_1884_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_1885_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_1886_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_1887_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B3,C3))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_1888_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
         => ( gbinomial(A,A3,K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A)),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A)),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_1889_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),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)) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2) ) ) ).

% neg_int_cases
tff(fact_1890_split__div_H,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,divide_divide(nat,M,Na))
    <=> ( ( ( Na = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q6: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q6)),M)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Q6)))
            & aa(nat,$o,P,Q6) ) ) ) ).

% split_div'
tff(fact_1891_ln__powr__bound,axiom,
    ! [Xa: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xa)),divide_divide(real,powr(real,Xa,A3),A3)) ) ) ).

% ln_powr_bound
tff(fact_1892_ln__powr__bound2,axiom,
    ! [Xa: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),Xa),A3)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A3,A3)),Xa)) ) ) ).

% ln_powr_bound2
tff(fact_1893_add__log__eq__powr,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Ya),aa(real,real,log(B3),Xa)) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B3,Ya)),Xa)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_1894_log__add__eq__powr,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B3),Xa)),Ya) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,B3,Ya))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_1895_minus__log__eq__powr,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,minus_minus(real,Ya),aa(real,real,log(B3),Xa)) = aa(real,real,log(B3),divide_divide(real,powr(real,B3,Ya),Xa)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_1896_root__sgn__power,axiom,
    ! [Na: nat,Ya: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,root(Na),aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Ya)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Ya)),Na))) = Ya ) ) ).

% root_sgn_power
tff(fact_1897_sgn__power__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,aa(real,real,root(Na),Xa))),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),aa(real,real,root(Na),Xa))),Na)) = Xa ) ) ).

% sgn_power_root
tff(fact_1898_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1899_sgn__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sgn_sgn(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_1900_zero__le__sgn__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sgn_sgn(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% zero_le_sgn_iff
tff(fact_1901_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: 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),Na)) = $ite(K = zero_zero(nat),zero_zero(nat),divide_divide(nat,M,Na)) ).

% nat_mult_div_cancel_disj
tff(fact_1902_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_1903_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(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),Na),K)) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,minus_minus(nat,Na),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_1904_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),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)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),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,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_less_add_iff2
tff(fact_1905_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),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)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_less_add_iff1
tff(fact_1906_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4) ) ).

% Compl_subset_Compl_iff
tff(fact_1907_Compl__anti__mono,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B2)),aa(set(A),set(A),uminus_uminus(set(A)),A4)) ) ).

% Compl_anti_mono
tff(fact_1908_Compl__disjoint,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = bot_bot(set(A)) ).

% Compl_disjoint
tff(fact_1909_Compl__disjoint2,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),A4) = bot_bot(set(A)) ).

% Compl_disjoint2
tff(fact_1910_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_1911_Diff__Compl,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),uminus_uminus(set(A)),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) ).

% Diff_Compl
tff(fact_1912_Compl__Diff__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),B2) ).

% Compl_Diff_eq
tff(fact_1913_subset__Compl__singleton,axiom,
    ! [A: $tType,A4: set(A),B3: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))))
    <=> ~ member(A,B3,A4) ) ).

% subset_Compl_singleton
tff(fact_1914_artanh__minus__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),Xa)) ) ) ).

% artanh_minus_real
tff(fact_1915_subset__Compl__self__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4))
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_1916_real__minus__mult__self__le,axiom,
    ! [U: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),U),U))),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Xa)) ).

% real_minus_mult_self_le
tff(fact_1917_Compl__Un,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B2)) ).

% Compl_Un
tff(fact_1918_Compl__Int,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),uminus_uminus(set(A)),B2)) ).

% Compl_Int
tff(fact_1919_Diff__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),B2)) ).

% Diff_eq
tff(fact_1920_disjoint__eq__subset__Compl,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_1921_Compl__insert,axiom,
    ! [A: $tType,Xa: A,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),uminus_uminus(set(A)),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_1922_real__add__less__0__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Ya)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),aa(real,real,uminus_uminus(real),Xa)) ) ).

% real_add_less_0_iff
tff(fact_1923_real__0__less__add__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),Xa)),Ya) ) ).

% real_0_less_add_iff
tff(fact_1924_real__0__le__add__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xa)),Ya) ) ).

% real_0_le_add_iff
tff(fact_1925_real__add__le__0__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Ya)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),aa(real,real,uminus_uminus(real),Xa)) ) ).

% real_add_le_0_iff
tff(fact_1926_abs__real__def,axiom,
    ! [A3: real] :
      aa(real,real,abs_abs(real),A3) = $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),zero_zero(real)),aa(real,real,uminus_uminus(real),A3),A3) ).

% abs_real_def
tff(fact_1927_zsgn__def,axiom,
    ! [I: int] :
      sgn_sgn(int,I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_1928_sgn__real__def,axiom,
    ! [A3: real] :
      sgn_sgn(real,A3) = $ite(
        A3 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_1929_powr__neg__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,uminus_uminus(real),one_one(real))) = divide_divide(real,one_one(real),Xa) ) ) ).

% powr_neg_one
tff(fact_1930_ln__add__one__self__le__self2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))),Xa) ) ).

% ln_add_one_self_le_self2
tff(fact_1931_ln__one__minus__pos__upper__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),Xa))),aa(real,real,uminus_uminus(real),Xa)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_1932_Bernoulli__inequality,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),Na)) ) ).

% Bernoulli_inequality
tff(fact_1933_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: 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),Na) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = Na ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1934_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( ( sgn_sgn(A,Xa) = zero_zero(A) )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_1935_log__minus__eq__powr,axiom,
    ! [B3: real,Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,minus_minus(real,aa(real,real,log(B3),Xa)),Ya) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),Xa),powr(real,B3,aa(real,real,uminus_uminus(real),Ya)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_1936_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na) )
      <=> ( M = Na ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1937_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ) ).

% nat_mult_less_cancel1
tff(fact_1938_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% nat_mult_le_cancel1
tff(fact_1939_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( 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),Na)) = divide_divide(nat,M,Na) ) ) ).

% nat_mult_div_cancel1
tff(fact_1940_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(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)),Na)) = aa(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,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_diff_add_eq2
tff(fact_1941_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(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)),Na)) = aa(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,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_diff_add_eq1
tff(fact_1942_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),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)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),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,minus_minus(nat,J),I)),U)),Na)) ) ) ).

% nat_le_add_iff2
tff(fact_1943_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),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)),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),Na) ) ) ).

% nat_le_add_iff1
tff(fact_1944_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),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)),Na) )
      <=> ( 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,minus_minus(nat,J),I)),U)),Na) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1945_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),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)),Na) )
      <=> ( 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,minus_minus(nat,I),J)),U)),M) = Na ) ) ) ).

% nat_eq_add_iff1
tff(fact_1946_ceiling__log__eq__powr__iff,axiom,
    ! [Xa: real,B3: real,K: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B3),Xa)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),powr(real,B3,aa(nat,real,semiring_1_of_nat(real),K))),Xa)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),powr(real,B3,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_1947_powr__int,axiom,
    ! [Xa: real,I: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(nat,real,power_power(real,Xa),aa(int,nat,nat2,I)),divide_divide(real,one_one(real),aa(nat,real,power_power(real,Xa),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_1948_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_1949_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,one_one(real)),divide_divide(real,Xa,aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),exp(real,aa(real,real,uminus_uminus(real),Xa))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_1950_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Na))),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,Xa,aa(nat,real,semiring_1_of_nat(real),Na)))),Na)),exp(real,Xa)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_1951_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( ( gcd_Gcd(A,A4) = zero_zero(A) )
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A)))) ) ) ).

% Gcd_0_iff
tff(fact_1952_ComplI,axiom,
    ! [A: $tType,C3: A,A4: set(A)] :
      ( ~ member(A,C3,A4)
     => member(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A4)) ) ).

% ComplI
tff(fact_1953_Compl__iff,axiom,
    ! [A: $tType,C3: A,A4: set(A)] :
      ( member(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A4))
    <=> ~ member(A,C3,A4) ) ).

% Compl_iff
tff(fact_1954_Compl__eq__Compl__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(set(A),set(A),uminus_uminus(set(A)),B2) )
    <=> ( A4 = B2 ) ) ).

% Compl_eq_Compl_iff
tff(fact_1955_exp__less__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),exp(real,Ya)) ) ).

% exp_less_mono
tff(fact_1956_exp__less__cancel__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),exp(real,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ).

% exp_less_cancel_iff
tff(fact_1957_exp__le__cancel__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),exp(real,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ).

% exp_le_cancel_iff
tff(fact_1958_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_1959_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_1960_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_1961_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_1962_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( exp(A,zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_1963_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W2),Z) ) ) ).

% of_int_le_iff
tff(fact_1964_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W2: int,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),W2)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),W2),Z) ) ) ).

% of_int_less_iff
tff(fact_1965_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_1966_Gcd__empty,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_empty
tff(fact_1967_one__less__exp__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% one_less_exp_iff
tff(fact_1968_exp__less__one__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_1969_exp__le__one__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_1970_one__le__exp__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),exp(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% one_le_exp_iff
tff(fact_1971_exp__ln__iff,axiom,
    ! [Xa: real] :
      ( ( exp(real,aa(real,real,ln_ln(real),Xa)) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% exp_ln_iff
tff(fact_1972_exp__ln,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( exp(real,aa(real,real,ln_ln(real),Xa)) = Xa ) ) ).

% exp_ln
tff(fact_1973_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_1974_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_1975_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_1976_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_1977_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_1978_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_1979_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_1980_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_1981_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_1982_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% zero_less_ceiling
tff(fact_1983_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_1984_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa) ) ) ).

% one_le_ceiling
tff(fact_1985_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_1986_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ).

% one_less_ceiling
tff(fact_1987_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: int,W2: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W2)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,B3),W2)),Xa) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_1988_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B3: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),aa(nat,int,power_power(int,B3),W2)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_1989_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: int,W2: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W2)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,B3),W2)),Xa) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_1990_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: int,B3: int,W2: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Xa)),aa(nat,A,power_power(A,aa(int,A,ring_1_of_int(A),B3)),W2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,power_power(int,B3),W2)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_1991_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),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_1992_nat__ceiling__le__eq,axiom,
    ! [Xa: real,A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,archimedean_ceiling(real,Xa))),A3)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),A3)) ) ).

% nat_ceiling_le_eq
tff(fact_1993_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_1994_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa) ) ) ).

% zero_le_ceiling
tff(fact_1995_ComplD,axiom,
    ! [A: $tType,C3: A,A4: set(A)] :
      ( member(A,C3,aa(set(A),set(A),uminus_uminus(set(A)),A4))
     => ~ member(A,C3,A4) ) ).

% ComplD
tff(fact_1996_double__complement,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = A4 ).

% double_complement
tff(fact_1997_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ).

% le_of_int_ceiling
tff(fact_1998_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A3))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),A3) ) ) ).

% ceiling_le
tff(fact_1999_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_2000_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),Xa) ) ) ).

% less_ceiling_iff
tff(fact_2001_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_le_of_int
tff(fact_2002_exp__less__cancel,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),exp(real,Ya))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ).

% exp_less_cancel
tff(fact_2003_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z2)),Xa) ) ).

% ex_of_int_less
tff(fact_2004_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z2: int] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_less_of_int
tff(fact_2005_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : exp(A,Xa) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_2006_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_2007_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,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_2008_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I4)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),aa(int,A,ring_1_of_int(A),I4)) )
             => aa(int,$o,P,I4) ) ) ) ).

% ceiling_split
tff(fact_2009_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A3: int] :
          ( ( archimedean_ceiling(A,Xa) = A3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),A3)),one_one(A))),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),A3)) ) ) ) ).

% ceiling_eq_iff
tff(fact_2010_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xa) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_2011_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))),one_one(A))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xa))) ) ) ).

% ceiling_correct
tff(fact_2012_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_2013_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa) ) ) ).

% le_ceiling_iff
tff(fact_2014_exp__total,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
     => ? [X3: real] : exp(real,X3) = Ya ) ).

% exp_total
tff(fact_2015_exp__gt__zero,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),exp(real,Xa)) ).

% exp_gt_zero
tff(fact_2016_not__exp__less__zero,axiom,
    ! [Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),exp(real,Xa)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_2017_not__exp__le__zero,axiom,
    ! [Xa: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_2018_exp__ge__zero,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),exp(real,Xa)) ).

% exp_ge_zero
tff(fact_2019_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Ya)),archimedean_ceiling(A,Xa)) ) ) ).

% ceiling_mono
tff(fact_2020_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Ya))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya) ) ) ).

% ceiling_less_cancel
tff(fact_2021_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q5)
         => aa(A,$o,aa(A,fun(A,$o),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,divide_divide(A,P3,Q5)))),Q5)) ) ) ).

% ceiling_divide_upper
tff(fact_2022_Gcd__int__greater__eq__0,axiom,
    ! [K4: set(int)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K4)) ).

% Gcd_int_greater_eq_0
tff(fact_2023_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,divide_divide(A,P3,Q5)))),one_one(A))),Q5)),P3) ) ) ).

% ceiling_divide_lower
tff(fact_2024_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Na)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Na)),one_one(A)))
           => ( archimedean_ceiling(A,Xa) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_2025_exp__gt__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),exp(real,Xa)) ) ).

% exp_gt_one
tff(fact_2026_exp__ge__add__one__self,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),exp(real,Xa)) ).

% exp_ge_add_one_self
tff(fact_2027_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R2),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archimedean_ceiling(A,R2)))) ) ).

% of_nat_ceiling
tff(fact_2028_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xa)),archimedean_ceiling(A,Ya))) ) ).

% ceiling_add_le
tff(fact_2029_real__nat__ceiling__ge,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(int,nat,nat2,archimedean_ceiling(real,Xa)))) ).

% real_nat_ceiling_ge
tff(fact_2030_real__of__int__div4,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),divide_divide(int,Na,Xa))),divide_divide(real,aa(int,real,ring_1_of_int(real),Na),aa(int,real,ring_1_of_int(real),Xa))) ).

% real_of_int_div4
tff(fact_2031_exp__ge__add__one__self__aux,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),exp(real,Xa)) ) ).

% exp_ge_add_one_self_aux
tff(fact_2032_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_2033_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),Xa)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ) ).

% of_int_leD
tff(fact_2034_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_pos
tff(fact_2035_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Na))),Xa)
         => ( ( Na = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% of_int_lessD
tff(fact_2036_lemma__exp__total,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Ya)
     => ? [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),aa(real,real,minus_minus(real,Ya),one_one(real)))
          & ( exp(real,X3) = Ya ) ) ) ).

% lemma_exp_total
tff(fact_2037_ln__ge__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),aa(real,real,ln_ln(real),Xa))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Ya)),Xa) ) ) ).

% ln_ge_iff
tff(fact_2038_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [X3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),X3)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int))))
          & ! [Y2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Y2)),Xa)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y2),one_one(int)))) )
             => ( Y2 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_2039_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
        ? [Z2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z2)),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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_2040_ln__x__over__x__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,aa(real,real,ln_ln(real),Ya),Ya)),divide_divide(real,aa(real,real,ln_ln(real),Xa),Xa)) ) ) ).

% ln_x_over_x_mono
tff(fact_2041_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,Xa: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(int,A,ring_1_of_int(A),Xa))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),Na)),Xa) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2042_int__le__real__less,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),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_2043_int__less__real__le,axiom,
    ! [Na: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Na),M)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real))),aa(int,real,ring_1_of_int(real),M)) ) ).

% int_less_real_le
tff(fact_2044_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xa: A,A3: A] :
          powr(A,Xa,A3) = $ite(Xa = zero_zero(A),zero_zero(A),exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,ln_ln(A),Xa)))) ) ).

% powr_def
tff(fact_2045_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B3))) ) ) ) ).

% mult_ceiling_le
tff(fact_2046_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(nat,A,power_power(A,exp(A,divide_divide(A,Xa,aa(nat,A,semiring_1_of_nat(A),Na)))),Na) = exp(A,Xa) ) ) ) ).

% exp_divide_power_eq
tff(fact_2047_real__of__int__div2,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,minus_minus(real,divide_divide(real,aa(int,real,ring_1_of_int(real),Na),aa(int,real,ring_1_of_int(real),Xa))),aa(int,real,ring_1_of_int(real),divide_divide(int,Na,Xa)))) ).

% real_of_int_div2
tff(fact_2048_real__of__int__div3,axiom,
    ! [Na: int,Xa: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,divide_divide(real,aa(int,real,ring_1_of_int(real),Na),aa(int,real,ring_1_of_int(real),Xa))),aa(int,real,ring_1_of_int(real),divide_divide(int,Na,Xa)))),one_one(real)) ).

% real_of_int_div3
tff(fact_2049_Gcd__remove0__nat,axiom,
    ! [M5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M5)
     => ( gcd_Gcd(nat,M5) = gcd_Gcd(nat,aa(set(nat),set(nat),minus_minus(set(nat),M5),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))))) ) ) ).

% Gcd_remove0_nat
tff(fact_2050_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,K))) ) ).

% of_int_of_nat
tff(fact_2051_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xa: A] : neg_numeral_dbl_dec(A,Xa) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Xa)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_2052_powr__real__of__int,axiom,
    ! [Xa: real,Na: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na),aa(nat,real,power_power(real,Xa),aa(int,nat,nat2,Na)),aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,Xa),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Na))))) ) ) ).

% powr_real_of_int
tff(fact_2053_floor__log__eq__powr__iff,axiom,
    ! [Xa: real,B3: real,K: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B3),Xa)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B3,aa(int,real,ring_1_of_int(real),K))),Xa)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),powr(real,B3,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_2054_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sinh(A,Xa) = zero_zero(A) )
        <=> member(A,exp(A,Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),one_one(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_2055_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B3)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_2056_power_Opower__eq__if,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),P3: A,M: nat] :
      power2(A,One,Times,P3,M) = $ite(M = zero_zero(nat),One,aa(A,A,aa(A,fun(A,A),Times,P3),power2(A,One,Times,P3,aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ).

% power.power_eq_if
tff(fact_2057_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q5)
         => aa(A,$o,aa(A,fun(A,$o),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,divide_divide(A,P3,Q5)))),one_one(A))),Q5)) ) ) ).

% floor_divide_upper
tff(fact_2058_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( rotate1(A,Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_2059_sinh__real__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xa)),sinh(real,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ).

% sinh_real_less_iff
tff(fact_2060_sinh__real__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),sinh(real,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ).

% sinh_real_le_iff
tff(fact_2061_sinh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sinh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% sinh_real_pos_iff
tff(fact_2062_sinh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_2063_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_2064_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_2065_sinh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sinh(real,Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% sinh_real_nonneg_iff
tff(fact_2066_sinh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_2067_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_2068_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_2069_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_2070_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_2071_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_2072_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_2073_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% inverse_positive_iff_positive
tff(fact_2074_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_2075_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_2076_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_2077_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_2078_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),A3)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_2079_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) ) ).

% zero_le_floor
tff(fact_2080_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_2081_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ).

% zero_less_floor
tff(fact_2082_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_2083_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ).

% one_le_floor
tff(fact_2084_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ).

% floor_less_one
tff(fact_2085_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_2086_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_2087_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B3) )
         => ( ( A3 != zero_zero(A) )
           => ( ( B3 != zero_zero(A) )
             => ( A3 = B3 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_2088_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
         => ( A3 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_2089_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_2090_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_2091_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Ya)) ) ) ).

% floor_mono
tff(fact_2092_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa) ) ).

% of_int_floor_le
tff(fact_2093_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% inverse_less_imp_less
tff(fact_2094_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less
tff(fact_2095_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_2096_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_2097_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_2098_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_2099_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_2100_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ).

% positive_imp_inverse_positive
tff(fact_2101_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Ya))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya) ) ) ).

% floor_less_cancel
tff(fact_2102_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_2103_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_2104_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,abs_abs(A),A3)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_2105_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archimedean_ceiling(A,Xa)) ) ).

% floor_le_ceiling
tff(fact_2106_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_2107_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B3)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_2108_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A3: A] : power2(A,One,Times,A3,zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_2109_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_2110_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_2111_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le
tff(fact_2112_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% inverse_le_imp_le
tff(fact_2113_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa) ) ) ) ).

% inverse_le_1_iff
tff(fact_2114_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_less_inverse
tff(fact_2115_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_2116_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_2117_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_2118_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% inverse_add
tff(fact_2119_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,minus_minus(A,B3),A3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_2120_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = divide_divide(A,one_one(A),A3) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_2121_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xa) ) ) ).

% le_floor_iff
tff(fact_2122_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_2123_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Ya))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya))) ) ).

% le_floor_add
tff(fact_2124_inverse__powr,axiom,
    ! [Ya: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
     => ( powr(real,aa(real,real,inverse_inverse(real),Ya),A3) = aa(real,real,inverse_inverse(real),powr(real,Ya,A3)) ) ) ).

% inverse_powr
tff(fact_2125_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_2126_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),aa(int,nat,nat2,archim6421214686448440834_floor(A,R2)))),R2) ) ) ).

% of_nat_floor
tff(fact_2127_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ) ) ).

% inverse_less_iff
tff(fact_2128_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ) ).

% inverse_le_iff
tff(fact_2129_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),Xa))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_2130_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),Xa)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa) ) ) ) ).

% inverse_less_1_iff
tff(fact_2131_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_le_inverse
tff(fact_2132_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,minus_minus(A,A3),B3))),aa(A,A,inverse_inverse(A),B3))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_2133_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),Xa) ) ) ).

% reals_Archimedean
tff(fact_2134_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(int,nat,nat2,archim6421214686448440834_floor(A,A3))),aa(int,nat,nat2,archim6421214686448440834_floor(A,B3)))),aa(int,nat,nat2,archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))) ) ).

% le_mult_nat_floor
tff(fact_2135_nat__floor__neg,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_2136_floor__eq3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = Na ) ) ) ).

% floor_eq3
tff(fact_2137_le__nat__floor,axiom,
    ! [Xa: nat,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xa)),A3)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(int,nat,nat2,archim6421214686448440834_floor(real,A3))) ) ).

% le_nat_floor
tff(fact_2138_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,archimedean_ceiling(A,Xa)),archim6421214686448440834_floor(A,Xa))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_2139_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_2140_floor__eq,axiom,
    ! [Na: int,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(int,real,ring_1_of_int(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xa) = Na ) ) ) ).

% floor_eq
tff(fact_2141_real__of__int__floor__add__one__ge,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_2142_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(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_2143_real__of__int__floor__ge__diff__one,axiom,
    ! [R2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,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_2144_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D6: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D6),E)
         => ( aa(real,$o,P,D6)
           => aa(real,$o,P,E) ) )
     => ( ! [N2: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono_1
tff(fact_2145_real__arch__inverse,axiom,
    ! [E2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
    <=> ? [N: nat] :
          ( ( N != zero_zero(nat) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N)))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N))),E2) ) ) ).

% real_arch_inverse
tff(fact_2146_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E2: real] :
      ( ! [D6: real,E: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),D6),E)
         => ( aa(real,$o,P,D6)
           => aa(real,$o,P,E) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N2))) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => aa(real,$o,P,E2) ) ) ) ).

% forall_pos_mono
tff(fact_2147_ln__inverse,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xa)) ) ) ).

% ln_inverse
tff(fact_2148_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_2149_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( ( Xa != zero_zero(A) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),Xa)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_2150_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xa)),one_one(A))
           => ( Xa = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_2151_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( member(A,Xa,ring_1_Ints(A))
         => ( member(A,Ya,ring_1_Ints(A))
           => ( ( Xa = Ya )
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xa),Ya))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_2152_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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,Xa) = Z ) ) ) ) ).

% floor_unique
tff(fact_2153_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,Xa) = A3 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_2154_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Ta))
        <=> ! [I4: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I4)),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
             => aa(int,$o,P,I4) ) ) ) ).

% floor_split
tff(fact_2155_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B3))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))) ) ) ) ).

% le_mult_floor
tff(fact_2156_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xa) ) ) ).

% less_floor_iff
tff(fact_2157_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),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_2158_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xa))),Xa)
          & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_2159_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => ? [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N2))),Xa) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_2160_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: nat,Na: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(nat,A,power_power(A,Xa),aa(nat,nat,minus_minus(nat,Na),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Xa),Na)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),Xa)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_2161_floor__eq4,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Na)))
       => ( aa(int,nat,nat2,archim6421214686448440834_floor(real,Xa)) = Na ) ) ) ).

% floor_eq4
tff(fact_2162_floor__eq2,axiom,
    ! [Na: int,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Na)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Na)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xa) = Na ) ) ) ).

% floor_eq2
tff(fact_2163_floor__divide__real__eq__div,axiom,
    ! [B3: int,A3: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B3)
     => ( archim6421214686448440834_floor(real,divide_divide(real,A3,aa(int,real,ring_1_of_int(real),B3))) = divide_divide(int,archim6421214686448440834_floor(real,A3),B3) ) ) ).

% floor_divide_real_eq_div
tff(fact_2164_log__inverse,axiom,
    ! [A3: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( aa(real,real,log(A3),aa(real,real,inverse_inverse(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A3),Xa)) ) ) ) ) ).

% log_inverse
tff(fact_2165_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q5: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Q5)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,divide_divide(A,P3,Q5)))),Q5)),P3) ) ) ).

% floor_divide_lower
tff(fact_2166_Cauchy__iff2,axiom,
    ! [X4: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X4)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
         => ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N)
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,X4,M2)),aa(nat,real,X4,N)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_2167_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Ya))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Ya)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xa)),archim6421214686448440834_floor(A,Ya))),one_one(int))) ) ).

% floor_add
tff(fact_2168_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,A3: A] :
          ( ( archimedean_frac(A,Xa) = A3 )
        <=> ( member(A,aa(A,A,minus_minus(A,Xa),A3),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_2169_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),Xa)) = $ite(member(A,Xa,ring_1_Ints(A)),zero_zero(A),aa(A,A,minus_minus(A,one_one(A)),archimedean_frac(A,Xa))) ) ).

% frac_neg
tff(fact_2170_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,Na,K)),modulo_modulo(int,Na,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_2171_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,divide_divide(int,Na,K)),modulo_modulo(int,Na,K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
              & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_2172_verit__le__mono__div__int,axiom,
    ! [A4: int,B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A4),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,
            aa(int,fun(int,$o),ord_less_eq(int),
              aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A4,Na)),
                $ite(modulo_modulo(int,B2,Na) = zero_zero(int),one_one(int),zero_zero(int)))),
            divide_divide(int,B2,Na)) ) ) ).

% verit_le_mono_div_int
tff(fact_2173_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_2174_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,A3) = zero_zero(A) ) ).

% mod_self
tff(fact_2175_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,zero_zero(A)) = A3 ) ).

% mod_by_0
tff(fact_2176_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,zero_zero(A),A3) = zero_zero(A) ) ).

% mod_0
tff(fact_2177_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3),B3) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_2178_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),B3) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_2179_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_2180_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : modulo_modulo(A,A3,one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_2181_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : divide_divide(A,modulo_modulo(A,A3,B3),B3) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_2182_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B3: A] : divide_divide(A,modulo_modulo(A,A3,B3),B3) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_2183_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_2184_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( archimedean_frac(A,Xa) = zero_zero(A) )
        <=> member(A,Xa,ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_2185_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : modulo_modulo(A,A3,aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2186_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),L)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2187_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),K)
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2188_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),archimedean_frac(A,Xa))
        <=> ~ member(A,Xa,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_2189_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),modulo_modulo(A,A3,B3)),A3) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2190_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,B3)),B3) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2191_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( ( modulo_modulo(A,A3,B3) = A3 )
        <=> ( divide_divide(A,A3,B3) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2192_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,M,K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_2193_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),modulo_modulo(int,K,L)) ) ).

% neg_mod_bound
tff(fact_2194_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,K,L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_2195_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,Xa)) ) ).

% frac_ge_0
tff(fact_2196_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),archimedean_frac(A,Xa)),one_one(A)) ) ).

% frac_lt_1
tff(fact_2197_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( modulo_modulo(A,A3,B3) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2198_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A3,B3)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2199_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_2200_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_2201_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( modulo_modulo(int,I,K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_2202_pos__mod__conj,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A3,B3))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),modulo_modulo(int,A3,B3)),B3) ) ) ).

% pos_mod_conj
tff(fact_2203_neg__mod__conj,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),modulo_modulo(int,A3,B3)),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),modulo_modulo(int,A3,B3)) ) ) ).

% neg_mod_conj
tff(fact_2204_zdiv__mono__strict,axiom,
    ! [A4: int,B2: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),A4),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => ( ( modulo_modulo(int,A4,Na) = zero_zero(int) )
         => ( ( modulo_modulo(int,B2,Na) = zero_zero(int) )
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,A4,Na)),divide_divide(int,B2,Na)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2205_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),modulo_modulo(int,K,L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_2206_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_2207_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),L),K)
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,minus_minus(int,K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_2208_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
         => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),modulo_modulo(A,divide_divide(A,A3,B3),C3))),modulo_modulo(A,A3,B3)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2209_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( ( archimedean_frac(A,Xa) = Xa )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_2210_int__mod__pos__eq,axiom,
    ! [A3: int,B3: int,Q5: int,R2: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),B3)
         => ( modulo_modulo(int,A3,B3) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_2211_int__mod__neg__eq,axiom,
    ! [A3: int,B3: int,Q5: int,R2: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R2) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),R2)
         => ( modulo_modulo(int,A3,B3) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_2212_split__zmod,axiom,
    ! [P: fun(int,$o),Na: int,K: int] :
      ( aa(int,$o,P,modulo_modulo(int,Na,K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,Na) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less(int),J3),K)
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Na = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_2213_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = aa(int,int,minus_minus(int,aa(int,int,minus_minus(int,L),one_one(int))),modulo_modulo(int,aa(int,int,minus_minus(int,K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2214_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Ya))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Ya)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xa)),archimedean_frac(A,Ya))),one_one(A))) ) ).

% frac_add
tff(fact_2215_zmod__minus1,axiom,
    ! [B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B3) = aa(int,int,minus_minus(int,B3),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2216_zmod__zmult2__eq,axiom,
    ! [C3: int,A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C3)
     => ( modulo_modulo(int,A3,aa(int,int,aa(int,fun(int,int),times_times(int),B3),C3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),modulo_modulo(int,divide_divide(int,A3,B3),C3))),modulo_modulo(int,A3,B3)) ) ) ).

% zmod_zmult2_eq
tff(fact_2217_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_2218_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          semiring_char_0_fact(A,M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_2219_verit__le__mono__div,axiom,
    ! [A4: nat,B2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(nat,$o,
            aa(nat,fun(nat,$o),ord_less_eq(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,A4,Na)),
                $ite(modulo_modulo(nat,B2,Na) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            divide_divide(nat,B2,Na)) ) ) ).

% verit_le_mono_div
tff(fact_2220_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2221_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B3,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2222_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W2: A,M: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,W2)),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,Z),M)),aa(nat,A,power_power(A,W2),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,minus_minus(A,Z),W2)))) ) ) ) ).

% norm_power_diff
tff(fact_2223_Gcd__fin__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = zero_zero(A) )
        <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))))
            & aa(set(A),$o,finite_finite2(A),A4) ) ) ) ).

% Gcd_fin_0_iff
tff(fact_2224_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: num,Na: num] :
          ( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),Na) )
        <=> ( M = Na ) ) ) ).

% numeral_eq_iff
tff(fact_2225_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ) ).

% numeral_le_iff
tff(fact_2226_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ) ).

% numeral_less_iff
tff(fact_2227_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V2: num,W2: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V2),W2))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_2228_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,Na: 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),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na)) ) ).

% numeral_times_numeral
tff(fact_2229_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V2: num,W2: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W2)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),W2))),Z) ) ).

% add_numeral_left
tff(fact_2230_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,Na: 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),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ) ).

% numeral_plus_numeral
tff(fact_2231_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_2232_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: 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),Na)) )
        <=> ( M = Na ) ) ) ).

% neg_numeral_eq_iff
tff(fact_2233_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),Na)) = aa(num,A,numeral_numeral(A),Na) ) ).

% of_nat_numeral
tff(fact_2234_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),Na) ) ).

% abs_numeral
tff(fact_2235_mod__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( modulo_modulo(nat,M,Na) = M ) ) ).

% mod_less
tff(fact_2236_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),M) ) ) ).

% neg_numeral_le_iff
tff(fact_2237_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V2: num,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C3)) ) ).

% distrib_left_numeral
tff(fact_2238_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B3: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(num,A,numeral_numeral(A),V2))) ) ).

% distrib_right_numeral
tff(fact_2239_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Na),M) ) ) ).

% neg_numeral_less_iff
tff(fact_2240_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V2: num,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,minus_minus(A,B3),C3)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V2)),C3)) ) ).

% right_diff_distrib_numeral
tff(fact_2241_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: A,B3: A,V2: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),aa(num,A,numeral_numeral(A),V2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V2))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(num,A,numeral_numeral(A),V2))) ) ).

% left_diff_distrib_numeral
tff(fact_2242_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: 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),Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2243_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: 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),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2244_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Na: 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),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Na)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2245_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: 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),Na))) = 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),Na))) ) ).

% add_neg_numeral_simps(3)
tff(fact_2246_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na))) ) ).

% diff_numeral_simps(3)
tff(fact_2247_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Na)) ) ).

% diff_numeral_simps(2)
tff(fact_2248_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),Na) ) ).

% abs_neg_numeral
tff(fact_2249_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2250_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( ( real_V7770717601297561774m_norm(A,Xa) = zero_zero(real) )
        <=> ( Xa = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2251_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2252_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_2253_numeral__less__real__of__nat__iff,axiom,
    ! [W2: num,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(num,real,numeral_numeral(real),W2)),aa(nat,real,semiring_1_of_nat(real),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),W2)),Na) ) ).

% numeral_less_real_of_nat_iff
tff(fact_2254_real__of__nat__less__numeral__iff,axiom,
    ! [Na: nat,W2: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(num,real,numeral_numeral(real),W2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),W2)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_2255_numeral__le__real__of__nat__iff,axiom,
    ! [Na: num,M: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Na)),aa(nat,real,semiring_1_of_nat(real),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Na)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_2256_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_2257_Gcd__fin_Oempty,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),bot_bot(set(A))) = zero_zero(A) ) ) ).

% Gcd_fin.empty
tff(fact_2258_Gcd__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = one_one(A) ) ) ) ).

% Gcd_fin.infinite
tff(fact_2259_Gcd__fin__eq__Gcd,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = gcd_Gcd(A,A4) ) ) ) ).

% Gcd_fin_eq_Gcd
tff(fact_2260_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B3) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_2261_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,aa(num,A,numeral_numeral(A),W2))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_2262_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,W2: num] :
          ( ( A3 = divide_divide(A,B3,aa(num,A,numeral_numeral(A),W2)) )
        <=> $ite(aa(num,A,numeral_numeral(A),W2) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_2263_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,W2: num,A3: A] :
          ( ( divide_divide(A,B3,aa(num,A,numeral_numeral(A),W2)) = A3 )
        <=> $ite(aa(num,A,numeral_numeral(A),W2) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2)),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_2264_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,aa(num,A,numeral_numeral(A),W2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))),B3) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_2265_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,aa(num,A,numeral_numeral(A),W2))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W2))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_2266_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),W2)) ) ).

% inverse_eq_divide_numeral
tff(fact_2267_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xa))
        <=> ( Xa != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2268_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xa)),zero_zero(real))
        <=> ( Xa = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2269_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_le_numeral_iff
tff(fact_2270_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Na)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_2271_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z),aa(num,int,numeral_numeral(int),Na)) ) ) ).

% of_int_less_numeral_iff
tff(fact_2272_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),Na)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_2273_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_2274_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V2)),Xa) ) ) ).

% numeral_le_floor
tff(fact_2275_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% floor_less_numeral
tff(fact_2276_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(num,A,numeral_numeral(A),V2)) ) ) ).

% ceiling_le_numeral
tff(fact_2277_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),V2)),Xa) ) ) ).

% numeral_less_ceiling
tff(fact_2278_powr__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(num,real,numeral_numeral(real),Na)) = aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),Na)) ) ) ).

% powr_numeral
tff(fact_2279_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B3) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2280_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2281_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,W2: num,A3: A] :
          ( ( divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = A3 )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2282_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,W2: num] :
          ( ( A3 = divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2283_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W2: num,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))),B3) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2284_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),divide_divide(A,B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2285_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_2286_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_2287_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) = divide_divide(A,one_one(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_2288_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,Xa: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(int,nat,nat2,A3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xa)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xa)),Na)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2289_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A3: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xa)),Na)),aa(int,nat,nat2,A3))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xa)),Na)),A3) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2290_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xa: num,Na: nat,A3: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xa)),Na)),aa(int,nat,nat2,A3))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xa)),Na)),A3) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2291_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,Xa: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(int,nat,nat2,A3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),Xa)),Na))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xa)),Na)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2292_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_2293_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)),Xa) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_2294_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: nat,I: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_2295_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Na: nat,Xa: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I)),Na)),aa(nat,A,semiring_1_of_nat(A),Xa))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I)),Na)),Xa) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_2296_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(num,int,numeral_numeral(int),V2)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))),Xa) ) ) ).

% numeral_less_floor
tff(fact_2297_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V2)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_2298_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),aa(num,int,numeral_numeral(int),V2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V2)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_2299_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V2)),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V2)),one_one(A))),Xa) ) ) ).

% numeral_le_ceiling
tff(fact_2300_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),Xa) ) ) ).

% neg_numeral_le_floor
tff(fact_2301_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archim6421214686448440834_floor(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% floor_less_neg_numeral
tff(fact_2302_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2303_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xa)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xa)),Na)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2304_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xa)),Na)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xa)),Na)),A3) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2305_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),Xa) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2306_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xa)),Na)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xa)),Na)),A3) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2307_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),Xa)),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),Xa)),Na)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2308_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),Xa) ) ) ).

% neg_numeral_less_floor
tff(fact_2309_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_2310_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,V2: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),archimedean_ceiling(A,Xa)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2311_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V2: num,Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V2))),archimedean_ceiling(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),one_one(A))),Xa) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2312_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2313_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)),A3) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2314_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: num,Na: nat,A3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)),A3) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2315_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xa: num,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa))),Na))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xa))),Na)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2316_fact__ge__self,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),semiring_char_0_fact(nat,Na)) ).

% fact_ge_self
tff(fact_2317_fact__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ).

% fact_mono_nat
tff(fact_2318_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [Na: nat] : semiring_char_0_fact(A,Na) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_2319_mod__less__eq__dividend,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,Na)),M) ).

% mod_less_eq_dividend
tff(fact_2320_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),Na) ) ).

% zero_neq_numeral
tff(fact_2321_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% numeral_neq_neg_numeral
tff(fact_2322_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Na: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),Na) ) ).

% neg_numeral_neq_numeral
tff(fact_2323_fact__less__mono__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Na)) ) ) ).

% fact_less_mono_nat
tff(fact_2324_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_zero
tff(fact_2325_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_gt_zero
tff(fact_2326_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,Na)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_2327_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_2328_mod__Suc,axiom,
    ! [M: nat,Na: nat] :
      modulo_modulo(nat,aa(nat,nat,suc,M),Na) = $ite(aa(nat,nat,suc,modulo_modulo(nat,M,Na)) = Na,zero_zero(nat),aa(nat,nat,suc,modulo_modulo(nat,M,Na))) ).

% mod_Suc
tff(fact_2329_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xa)) ) ).

% norm_ge_zero
tff(fact_2330_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Na)) ) ).

% fact_ge_1
tff(fact_2331_mod__induct,axiom,
    ! [P: fun(nat,$o),Na: nat,P3: nat,M: nat] :
      ( aa(nat,$o,P,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),P3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),P3)
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),P3)
               => ( aa(nat,$o,P,N2)
                 => aa(nat,$o,P,modulo_modulo(nat,aa(nat,nat,suc,N2),P3)) ) )
           => aa(nat,$o,P,M) ) ) ) ) ).

% mod_induct
tff(fact_2332_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Na: nat] :
      ( ! [M4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M4),zero_zero(nat))
     => ( ! [M4: nat,N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N2),modulo_modulo(nat,M4,N2))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M4),N2) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Na) ) ) ).

% gcd_nat_induct
tff(fact_2333_mod__less__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),modulo_modulo(nat,M,Na)),Na) ) ).

% mod_less_divisor
tff(fact_2334_mod__Suc__le__divisor,axiom,
    ! [M: nat,Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,aa(nat,nat,suc,Na))),Na) ).

% mod_Suc_le_divisor
tff(fact_2335_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_2336_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_le_numeral
tff(fact_2337_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_less_numeral
tff(fact_2338_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_2339_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ).

% fact_mono
tff(fact_2340_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) ) ).

% one_le_numeral
tff(fact_2341_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( modulo_modulo(nat,M,D2) = zero_zero(nat) )
     => ? [Q3: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q3) ) ).

% mod_eq_0D
tff(fact_2342_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),Na)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_2343_mod__geq,axiom,
    ! [M: nat,Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
     => ( modulo_modulo(nat,M,Na) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Na),Na) ) ) ).

% mod_geq
tff(fact_2344_mod__if,axiom,
    ! [M: nat,Na: nat] :
      modulo_modulo(nat,M,Na) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na),M,modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Na),Na)) ).

% mod_if
tff(fact_2345_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2346_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) ) ).

% neg_numeral_le_numeral
tff(fact_2347_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% zero_neq_neg_numeral
tff(fact_2348_le__mod__geq,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( modulo_modulo(nat,M,Na) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),Na),Na) ) ) ).

% le_mod_geq
tff(fact_2349_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2350_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Na)) ) ).

% neg_numeral_less_numeral
tff(fact_2351_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Xa)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xa)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_2352_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),Na) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_2353_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) ) ).

% one_neq_neg_numeral
tff(fact_2354_fact__ge__Suc__0__nat,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Na)) ).

% fact_ge_Suc_0_nat
tff(fact_2355_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,divide_divide(A,A3,B3)) = divide_divide(real,real_V7770717601297561774m_norm(A,A3),real_V7770717601297561774m_norm(A,B3)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2356_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,Na: nat,Z: A] :
          ( ( aa(nat,A,power_power(A,W2),Na) = aa(nat,A,power_power(A,Z),Na) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( real_V7770717601297561774m_norm(A,W2) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2357_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A,R2: real,Ya: A,S3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Ya)),S3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Ya))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S3)) ) ) ) ).

% norm_mult_less
tff(fact_2358_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xa: A,Ya: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Ya))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Ya))) ) ).

% norm_mult_ineq
tff(fact_2359_mod__le__divisor,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),modulo_modulo(nat,M,Na)),Na) ) ).

% mod_le_divisor
tff(fact_2360_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,R2: real,Ya: A,S3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Ya)),S3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S3)) ) ) ) ).

% norm_add_less
tff(fact_2361_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Ya: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Ya))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya))),E2) ) ) ).

% norm_triangle_lt
tff(fact_2362_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,R2: real,B3: A,S3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A3)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B3)),S3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S3)) ) ) ) ).

% norm_triangle_mono
tff(fact_2363_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Ya: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Ya))) ) ).

% norm_triangle_ineq
tff(fact_2364_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Ya: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Ya))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya))),E2) ) ) ).

% norm_triangle_le
tff(fact_2365_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B3: A,C3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))),C3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B3)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A3)),C3)) ) ) ).

% norm_add_leD
tff(fact_2366_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A,Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,power_power(A,Xa),Na))),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,Xa)),Na)) ) ).

% norm_power_ineq
tff(fact_2367_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Ya: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Ya))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Ya),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2368_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Ya: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xa)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Ya)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Ya)))) ) ).

% norm_triangle_sub
tff(fact_2369_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A3),B3))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B3))) ) ).

% norm_triangle_ineq4
tff(fact_2370_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Ya: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Ya))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Ya),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_le
tff(fact_2371_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A,Ya: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,Ya))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xa),Ya))),E2) ) ) ).

% norm_triangle_le_diff
tff(fact_2372_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_zero_le_neg_numeral
tff(fact_2373_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_2374_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Na)) ) ) ) ).

% fact_less_mono
tff(fact_2375_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_2376_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) ) ).

% not_zero_less_neg_numeral
tff(fact_2377_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B3))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))) ) ).

% norm_diff_ineq
tff(fact_2378_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B3: A,C3: A] :
          ( ( aa(num,A,numeral_numeral(A),W2) = divide_divide(A,B3,C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3) = B3,aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_2379_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C3: A,W2: num] :
          ( ( divide_divide(A,B3,C3) = aa(num,A,numeral_numeral(A),W2) )
        <=> $ite(C3 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3),aa(num,A,numeral_numeral(A),W2) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_2380_div__less__mono,axiom,
    ! [A4: nat,B2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A4),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( modulo_modulo(nat,A4,Na) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B2,Na) = zero_zero(nat) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),divide_divide(nat,A4,Na)),divide_divide(nat,B2,Na)) ) ) ) ) ).

% div_less_mono
tff(fact_2381_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B3))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A3),B3))) ) ).

% norm_triangle_ineq2
tff(fact_2382_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_le_neg_numeral
tff(fact_2383_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),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_2384_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_2385_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_le_numeral
tff(fact_2386_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_2387_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2388_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_less_neg_numeral
tff(fact_2389_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(A,$o),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_2390_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_less_numeral
tff(fact_2391_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(A,$o),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_2392_mod__eq__nat1E,axiom,
    ! [M: nat,Q5: nat,Na: nat] :
      ( ( modulo_modulo(nat,M,Q5) = modulo_modulo(nat,Na,Q5) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
       => ~ ! [S4: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),S4)) ) ) ).

% mod_eq_nat1E
tff(fact_2393_mod__eq__nat2E,axiom,
    ! [M: nat,Q5: nat,Na: nat] :
      ( ( modulo_modulo(nat,M,Q5) = modulo_modulo(nat,Na,Q5) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ~ ! [S4: nat] : Na != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q5),S4)) ) ) ).

% mod_eq_nat2E
tff(fact_2394_nat__mod__eq__lemma,axiom,
    ! [Xa: nat,Na: nat,Ya: nat] :
      ( ( modulo_modulo(nat,Xa,Na) = modulo_modulo(nat,Ya,Na) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ya),Xa)
       => ? [Q3: nat] : Xa = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Ya),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_2395_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( modulo_modulo(A,semiring_char_0_fact(A,Na),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_2396_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A3)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A3)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_2397_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Na)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,Na),Na))) ) ).

% fact_le_power
tff(fact_2398_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Xa))),exp(real,real_V7770717601297561774m_norm(A,Xa))) ) ).

% norm_exp
tff(fact_2399_powr__neg__numeral,axiom,
    ! [Xa: real,Na: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),Na))) = divide_divide(real,one_one(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),Na))) ) ) ).

% powr_neg_numeral
tff(fact_2400_fact__diff__Suc,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,suc,M))
     => ( semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Na)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,M),Na))) ) ) ).

% fact_diff_Suc
tff(fact_2401_fact__div__fact__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,semiring_char_0_fact(nat,Na),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Na),R2)))),aa(nat,nat,power_power(nat,Na),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_2402_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W2: A,Na: nat] :
          ( ( aa(nat,A,power_power(A,W2),Na) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W2) = one_one(real) )
            | ( Na = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2403_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B3: A,C3: A,D2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C3),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A3),C3))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,B3),D2)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2404_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] :
          real_V7770717601297561774m_norm(A,sgn_sgn(A,Xa)) = $ite(Xa = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_2405_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C3)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_2406_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B3,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_2407_split__mod,axiom,
    ! [P: fun(nat,$o),M: nat,Na: nat] :
      ( aa(nat,$o,P,modulo_modulo(nat,M,Na))
    <=> ( ( ( Na = zero_zero(nat) )
         => aa(nat,$o,P,M) )
        & ( ( Na != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),Na)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),I4)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2408_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C3: A,W2: num] :
          ( ( divide_divide(A,B3,C3) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) )
        <=> $ite(C3 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2409_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W2: num,B3: A,C3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = divide_divide(A,B3,C3) )
        <=> $ite(C3 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3) = B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2410_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A,B3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B3)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A3),B3))) ) ).

% norm_triangle_ineq3
tff(fact_2411_nat__mod__distrib,axiom,
    ! [Xa: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
       => ( aa(int,nat,nat2,modulo_modulo(int,Xa,Ya)) = modulo_modulo(nat,aa(int,nat,nat2,Xa),aa(int,nat,nat2,Ya)) ) ) ) ).

% nat_mod_distrib
tff(fact_2412_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X4: fun(A,B)] :
          ( ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & ! [N: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X4,N))),K5) )
        <=> ? [N6: nat] :
            ! [N: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X4,N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% lemma_NBseq_def
tff(fact_2413_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X4: fun(A,B)] :
          ( ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & ! [N: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X4,N))),K5) )
        <=> ? [N6: nat] :
            ! [N: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X4,N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% lemma_NBseq_def2
tff(fact_2414_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),divide_divide(A,B3,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W2)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_2415_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,B3,C3)),aa(num,A,numeral_numeral(A),W2))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W2)),C3)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W2))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_2416_Suc__times__mod__eq,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2417_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W2: num,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),divide_divide(A,B3,C3))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B3),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2418_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),divide_divide(A,B3,C3)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> $ite(
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
              aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),
              $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2))),C3)),B3),aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2419_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R2),real_V7770717601297561774m_norm(A,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xa))),aa(real,real,inverse_inverse(real),R2)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_2420_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M9: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
               => ! [N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X4,M3)),aa(nat,A,X4,N3)))),E2) ) ) ) ) ) ).

% CauchyD
tff(fact_2421_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X4,M4)),aa(nat,A,X4,N2)))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X4) ) ) ).

% CauchyI
tff(fact_2422_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X4)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M8: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X4,M2)),aa(nat,A,X4,N)))),E3) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_2423_assms_I2_J,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),x),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),n)) ).

% assms(2)
tff(fact_2424_enat__ord__number_I1_J,axiom,
    ! [M: num,Na: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(1)
tff(fact_2425_enat__ord__number_I2_J,axiom,
    ! [M: num,Na: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)) ) ).

% enat_ord_number(2)
tff(fact_2426_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K4: real,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K4)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K4)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Na)),aa(nat,A,power_power(A,Z),Na)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,Z),aa(nat,nat,minus_minus(nat,Na),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),Na)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,power_power(real,K4),aa(nat,nat,minus_minus(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2427_bounded__linear__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V4916620083959148203axioms(A,B,F2)
        <=> ? [K5: real] :
            ! [X: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),K5)) ) ) ).

% bounded_linear_axioms_def
tff(fact_2428_bounded__linear__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( ? [K6: real] :
            ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),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)),K6))
         => real_V4916620083959148203axioms(A,B,F2) ) ) ).

% bounded_linear_axioms.intro
tff(fact_2429_i0__less,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na)
    <=> ( Na != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_2430_pow__sum,axiom,
    ! [A3: nat,B3: nat] : divide_divide(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B3)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3) ).

% pow_sum
tff(fact_2431_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xa: nat,Na: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xa)
     => ( vEBT_invar_vebt(Tree,Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ).

% member_bound
tff(fact_2432_valid__pres__insert,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => vEBT_invar_vebt(vEBT_vebt_insert(Ta,Xa),Na) ) ) ).

% valid_pres_insert
tff(fact_2433_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat,Ya: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ya),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xa)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Ya)),Xa) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_2434_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xa)),Xa) ) ) ).

% valid_insert_both_member_options_add
tff(fact_2435_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Na: nat,Xa: nat,Ya: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ya),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
         => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(Ta,Xa)),Ya)
           => ( aa(nat,$o,vEBT_vebt_member(Ta),Ya)
              | ( Xa = Ya ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_2436_semiring__norm_I78_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(78)
tff(fact_2437_semiring__norm_I71_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(71)
tff(fact_2438_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),one2) ).

% semiring_norm(75)
tff(fact_2439_semiring__norm_I68_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),one2),Na) ).

% semiring_norm(68)
tff(fact_2440_bit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H,L,D2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),D2))),L) ).

% bit_concat_def
tff(fact_2441_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] :
          ( ( aa(num,A,numeral_numeral(A),Na) = one_one(A) )
        <=> ( Na = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_2442_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Na) )
        <=> ( one2 = Na ) ) ) ).

% one_eq_numeral_iff
tff(fact_2443_num__double,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),Na) = bit0(Na) ).

% num_double
tff(fact_2444_semiring__norm_I76_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit0(Na)) ).

% semiring_norm(76)
tff(fact_2445_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_2446_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) )
        <=> ( Na = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2447_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Na: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( Na = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2448_Suc__numeral,axiom,
    ! [Na: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Na)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ).

% Suc_numeral
tff(fact_2449_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2450_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),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_2451_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),bit0(one2)) ) ) ).

% one_add_one
tff(fact_2452_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_2453_add__2__eq__Suc_H,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,Na)) ).

% add_2_eq_Suc'
tff(fact_2454_add__2__eq__Suc,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) = aa(nat,nat,suc,aa(nat,nat,suc,Na)) ).

% add_2_eq_Suc
tff(fact_2455_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_2456_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Na)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2)) ) ).

% numeral_plus_one
tff(fact_2457_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Na)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Na)) ) ).

% one_plus_numeral
tff(fact_2458_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Na)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),one2) ) ) ).

% numeral_le_one_iff
tff(fact_2459_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Na: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),Na))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),Na) ) ) ).

% one_less_numeral_iff
tff(fact_2460_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_2461_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_2462_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
           => ( ( aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( Xa = Ya ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_2463_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_2464_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),bit0(one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_2465_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_2466_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(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),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_2467_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_2468_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Ya = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_2469_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2470_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2471_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Na: nat] :
      ( ( modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2472_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Na))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Na)) ) ).

% diff_numeral_special(3)
tff(fact_2473_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(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_2474_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),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_2475_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_2476_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_2477_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,Xa))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa) ) ) ).

% one_less_floor
tff(fact_2478_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_2479_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2480_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Ya: extended_enat,Xa: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Z),Ya)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xa),aa(extended_enat,extended_enat,minus_minus(extended_enat,Ya),Z)) = aa(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),Xa),Ya)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_2481_ile0__eq,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Na),zero_zero(extended_enat))
    <=> ( Na = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_2482_i0__lb,axiom,
    ! [Na: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),zero_zero(extended_enat)),Na) ).

% i0_lb
tff(fact_2483_le__num__One__iff,axiom,
    ! [Xa: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Xa),one2)
    <=> ( Xa = one2 ) ) ).

% le_num_One_iff
tff(fact_2484_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),Na: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M3: extended_enat] :
              ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M3),N2)
             => aa(extended_enat,$o,P,M3) )
         => aa(extended_enat,$o,P,N2) )
     => aa(extended_enat,$o,P,Na) ) ).

% enat_less_induct
tff(fact_2485_not__iless0,axiom,
    ! [Na: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_2486_add__One__commute,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Na) = aa(num,num,aa(num,fun(num,num),plus_plus(num),Na),one2) ).

% add_One_commute
tff(fact_2487_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Na))
    <=> ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),M)
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na) ) ) ).

% enat_0_less_mult_iff
tff(fact_2488_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_2489_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),bit0(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_2490_pos2,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_2491_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),bit0(one2)) ).

% nat_1_add_1
tff(fact_2492_less__exp,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% less_exp
tff(fact_2493_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,K),M)) ) ).

% self_le_ge2_pow
tff(fact_2494_power2__nat__le__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,power_power(nat,Na),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_eq_le
tff(fact_2495_power2__nat__le__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% power2_nat_le_imp_le
tff(fact_2496_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_2497_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_2498_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ) ).

% power2_le_imp_le
tff(fact_2499_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
             => ( Xa = Ya ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_2500_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_2501_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_2502_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = 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),bit0(one2))),A3)),B3) ) ).

% left_add_twice
tff(fact_2503_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),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2_right
tff(fact_2504_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),bit0(one2))),Z) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2
tff(fact_2505_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),aa(A,A,abs_abs(A),Ya))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_le_square_iff
tff(fact_2506_less__2__cases,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( ( Na = zero_zero(nat) )
        | ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_2507_less__2__cases__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))
    <=> ( ( Na = zero_zero(nat) )
        | ( Na = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_2508_card__2__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
    <=> ? [X: A,Y4: A] :
          ( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y4),bot_bot(set(A)))) )
          & ( X != Y4 ) ) ) ).

% card_2_iff
tff(fact_2509_nat__2,axiom,
    aa(int,nat,nat2,aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_2510_nat__induct2,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N2: nat] :
              ( aa(nat,$o,P,N2)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_induct2
tff(fact_2511_two__realpow__ge__one,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(one2))),Na)) ).

% two_realpow_ge_one
tff(fact_2512_square__fact__le__2__fact,axiom,
    ! [Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,Na)),semiring_char_0_fact(real,Na))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% square_fact_le_2_fact
tff(fact_2513_realpow__square__minus__le,axiom,
    ! [U: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,power_power(real,U),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_2514_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Na)),aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,K),M)),aa(nat,nat,power_power(nat,K),Na))) ) ).

% diff_le_diff_pow
tff(fact_2515_ln__2__less__1,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(one2)))),one_one(real)) ).

% ln_2_less_1
tff(fact_2516_not__exp__less__eq__0__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_2517_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya) ) ) ) ).

% power2_less_imp_less
tff(fact_2518_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_2519_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ).

% half_gt_zero_iff
tff(fact_2520_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))
        <=> ( ( Xa = zero_zero(A) )
            & ( Ya = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_2521_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_power2_ge_zero
tff(fact_2522_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_2523_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( ( Xa != zero_zero(A) )
            | ( Ya != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_2524_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_2525_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_2526_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),Ya) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2527_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% of_nat_less_two_power
tff(fact_2528_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_2529_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_2530_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ) ).

% zero_le_even_power'
tff(fact_2531_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,M: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_2532_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xa)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_2533_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),Xa)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_2534_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,P,N2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa(nat,$o,P,N2)
             => aa(nat,$o,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))) )
         => aa(nat,$o,P,Na) ) ) ) ).

% nat_bit_induct
tff(fact_2535_div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_2536_Suc__n__div__2__gt__zero,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),divide_divide(nat,aa(nat,nat,suc,Na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_2537_L2__set__mult__ineq__lemma,axiom,
    ! [A3: real,C3: real,B3: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A3),C3))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),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,power_power(real,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,D2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,C3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_2538_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bit0(Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Na)),aa(num,A,numeral_numeral(A),Na)) ) ).

% numeral_Bit0
tff(fact_2539_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% exp_half_le2
tff(fact_2540_exp__plus__inverse__exp,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa)))) ).

% exp_plus_inverse_exp
tff(fact_2541_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A3) = A3 ) ).

% mult_numeral_1
tff(fact_2542_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% mult_numeral_1_right
tff(fact_2543_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_2544_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] : divide_divide(A,A3,aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% divide_numeral_1
tff(fact_2545_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_2546_Suc__nat__number__of__add,axiom,
    ! [V2: num,Na: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Na)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V2),one2))),Na) ).

% Suc_nat_number_of_add
tff(fact_2547_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_2548_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)),Ya)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,power_power(A,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_squares_bound
tff(fact_2549_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3)
           => ( modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)) = modulo_modulo(A,A3,B3) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2550_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ( divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2551_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_2552_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A))
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_2553_ex__power__ivl1,axiom,
    ! [B3: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B3),N2)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_2554_ex__power__ivl2,axiom,
    ! [B3: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B3),N2)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_2555_plus__inverse__ge__2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,inverse_inverse(real),Xa))) ) ).

% plus_inverse_ge_2
tff(fact_2556_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_2557_less__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_2558_le__log2__of__power,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_2559_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))) = divide_divide(A,A3,B3) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2560_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A3,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2561_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num,Q5: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit0(Na)),aa(num,A,numeral_numeral(A),bit0(Q5))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),Q5)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2562_num_Osize_I5_J,axiom,
    ! [X23: num] : aa(num,nat,size_size(num),bit0(X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_2563_log2__of__power__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_less
tff(fact_2564_exp__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),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)),Xa)),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_2565_pos__zdiv__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( 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),bit0(one2))),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = divide_divide(int,B3,A3) ) ) ).

% pos_zdiv_mult_2
tff(fact_2566_neg__zdiv__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int))
     => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = divide_divide(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B3),one_one(int)),A3) ) ) ).

% neg_zdiv_mult_2
tff(fact_2567_pos__zmod__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3)
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,B3,A3))) ) ) ).

% pos_zmod_mult_2
tff(fact_2568_real__le__x__sinh,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,aa(real,real,minus_minus(real,exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_2569_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B3: 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))),B3) = aa(A,A,uminus_uminus(A),B3) ) ).

% mult_1s_ring_1(1)
tff(fact_2570_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B3) ) ).

% mult_1s_ring_1(2)
tff(fact_2571_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_2572_real__le__abs__sinh,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,abs_abs(real),divide_divide(real,aa(real,real,minus_minus(real,exp(real,Xa)),aa(real,real,inverse_inverse(real),exp(real,Xa))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_2573_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_2574_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xa: A,Ya: A] :
          ( ( aa(nat,A,power_power(A,U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xa),Ya) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Ya)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_2575_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => ( ( modulo_modulo(A,Xa,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = modulo_modulo(A,Xa,M) )
              | ( modulo_modulo(A,Xa,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,Xa,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2576_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))
             => ( aa(A,A,minus_minus(A,modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3) = modulo_modulo(A,A3,B3) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2577_log2__of__power__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Na)) ) ) ).

% log2_of_power_le
tff(fact_2578_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z)))) ) ) ).

% exp_bound_lemma
tff(fact_2579_real__exp__bound__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xa)),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),bit0(one2))),Xa))) ) ) ).

% real_exp_bound_lemma
tff(fact_2580_exp__lower__Taylor__quadratic,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),divide_divide(real,aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,real,numeral_numeral(real),bit0(one2))))),exp(real,Xa)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2581_ln__one__plus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,Xa),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2582_neg__zmod__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int))
     => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),modulo_modulo(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B3),one_one(int)),A3))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2583_floor__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_2584_sinh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( sinh(real,aa(real,real,ln_ln(real),Xa)) = divide_divide(real,aa(real,real,minus_minus(real,Xa),aa(real,real,inverse_inverse(real),Xa)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_2585_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(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)),Xa))),Xa))),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2586_arctan__double,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),arctan(Xa)) = arctan(divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xa),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_2587_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),modulo_modulo(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))
             => ( 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),bit0(one2))),divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))),one_one(A)) = divide_divide(A,A3,B3) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2588_ln__one__minus__pos__lower__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,uminus_uminus(real),Xa)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),Xa))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2589_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(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)),Xa))),Xa))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2590_floor__log__nat__eq__if,axiom,
    ! [B3: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B3),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Na) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_2591_floor__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Na) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,B3),Na)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_2592_ceiling__log2__div2,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_2593_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(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)),Xa))),Xa))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2594_ceiling__log__nat__eq__if,axiom,
    ! [B3: nat,Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B3),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),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),Na)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2595_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),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),Na)),one_one(int)) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,power_power(nat,B3),Na)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_2596_inrange,axiom,
    ! [Ta: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(Ta,Na)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(Ta)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat)))) ) ).

% inrange
tff(fact_2597_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xa: A] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X3)
             => aa(A,$o,aa(A,fun(A,$o),P,X3),aa(nat,A,power_power(A,X3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P,aa(A,A,abs_abs(A),Xa)),aa(nat,A,power_power(A,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_2598_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_2599_low__inv,axiom,
    ! [Xa: nat,Na: nat,Ya: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( 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),Ya),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Xa),Na) = Xa ) ) ).

% low_inv
tff(fact_2600_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X3,Na) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na) ) ) ).

% set_n_deg_not_0
tff(fact_2601_high__inv,axiom,
    ! [Xa: nat,Na: nat,Ya: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( 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),Ya),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),Xa),Na) = Ya ) ) ).

% high_inv
tff(fact_2602_bit__split__inv,axiom,
    ! [Xa: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xa,D2),vEBT_VEBT_low(Xa,D2),D2) = Xa ).

% bit_split_inv
tff(fact_2603_high__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_high(Xa,Na) = divide_divide(nat,Xa,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% high_def
tff(fact_2604_low__def,axiom,
    ! [Xa: nat,Na: nat] : vEBT_VEBT_low(Xa,Na) = modulo_modulo(nat,Xa,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% low_def
tff(fact_2605_high__bound__aux,axiom,
    ! [Ma: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Ma,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% high_bound_aux
tff(fact_2606_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L2: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L2,H2) )
        <=> ( ( ( L = L2 )
              & ( H = H2 ) )
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
              & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L2),H2) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_2607_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_2608_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(A),$o,finite_finite2(A),aa(list(A),set(A),set2(A),Xs)) ).

% List.finite_set
tff(fact_2609_set__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se5668285175392031749et_bit(int,Na,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_2610_set__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se5668285175392031749et_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_2611_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or1337092689740270186AtMost(nat,L,U)) ).

% finite_atLeastAtMost
tff(fact_2612_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B3) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_2613_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% atLeastatMost_empty_iff
tff(fact_2614_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_2615_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_2616_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% infinite_Icc_iff
tff(fact_2617_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C3),bot_bot(set(A))) )
        <=> ( ( A3 = B3 )
            & ( B3 = C3 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_2618_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : set_or1337092689740270186AtMost(A,A3,A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_2619_finite__list,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A4 ) ).

% finite_list
tff(fact_2620_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B2)
    <=> ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => member(A,X,B2) ) ) ).

% subset_code(1)
tff(fact_2621_set__bit__greater__eq,axiom,
    ! [K: int,Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),bit_se5668285175392031749et_bit(int,Na,K)) ).

% set_bit_greater_eq
tff(fact_2622_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(set(A),$o,finite_finite2(A),set_or1337092689740270186AtMost(A,A3,B3)) ) ) ).

% infinite_Icc
tff(fact_2623_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_2624_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
         => ( set_or1337092689740270186AtMost(A,A3,B3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_2625_all__nat__less,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),Na)
         => aa(nat,$o,P,M2) )
    <=> ! [X: nat] :
          ( member(nat,X,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less
tff(fact_2626_ex__nat__less,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),Na)
          & aa(nat,$o,P,M2) )
    <=> ? [X: nat] :
          ( member(nat,X,set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less
tff(fact_2627_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2)
                & ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A3)
                  | aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),D2) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_2628_length__pos__if__in__set,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_2629_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(list(A),set(A),set2(A),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% card_length
tff(fact_2630_atLeast0__atMost__Suc,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ).

% atLeast0_atMost_Suc
tff(fact_2631_atLeastAtMost__insertL,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = set_or1337092689740270186AtMost(nat,M,Na) ) ) ).

% atLeastAtMost_insertL
tff(fact_2632_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,Na)),set_or1337092689740270186AtMost(nat,M,Na)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2633_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( set_or1337092689740270186AtMost(nat,M,Na) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2634_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N4),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))
     => aa(set(nat),$o,finite_finite2(nat),N4) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_2635_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_2636_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xa: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_low(Xa,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_2637_complex__mod__minus__le__complex__mod,axiom,
    ! [Xa: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Xa))),real_V7770717601297561774m_norm(complex,Xa)) ).

% complex_mod_minus_le_complex_mod
tff(fact_2638_complex__mod__triangle__ineq2,axiom,
    ! [B3: complex,A3: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),B3),A3))),real_V7770717601297561774m_norm(complex,B3))),real_V7770717601297561774m_norm(complex,A3)) ).

% complex_mod_triangle_ineq2
tff(fact_2639_set__encode__insert,axiom,
    ! [A4: set(nat),Na: nat] :
      ( aa(set(nat),$o,finite_finite2(nat),A4)
     => ( ~ member(nat,Na,A4)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Na),A4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(set(nat),nat,nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_2640_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_2641_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat,A3: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A3) = $ite(Na = zero_zero(nat),aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_2642_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),set(A),set2(A),union(A,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ).

% set_union
tff(fact_2643_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Xa),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Ya))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Ya)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,Xa) = Ya ) ) ) ) ).

% round_unique
tff(fact_2644_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),bit0(one2))) ) ) ).

% dbl_simps(4)
tff(fact_2645_unset__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2638667681897837118et_bit(int,Na,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2646_unset__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2638667681897837118et_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_2647_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_2648_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_2649_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2650_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),bit0(K)) ) ).

% dbl_simps(5)
tff(fact_2651_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_2652_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_2653_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% dbl_simps(3)
tff(fact_2654_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A3) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_2655_unset__bit__less__eq,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_se2638667681897837118et_bit(int,Na,K)),K) ).

% unset_bit_less_eq
tff(fact_2656_set__encode__eq,axiom,
    ! [A4: set(nat),B2: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A4)
     => ( aa(set(nat),$o,finite_finite2(nat),B2)
       => ( ( aa(set(nat),nat,nat_set_encode,A4) = aa(set(nat),nat,nat_set_encode,B2) )
        <=> ( A4 = B2 ) ) ) ) ).

% set_encode_eq
tff(fact_2657_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xa: A] : neg_numeral_dbl(A,Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Xa) ) ).

% dbl_def
tff(fact_2658_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Na)),set_or1337092689740270186AtMost(int,M,Na)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2659_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),bot_bot(set(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert2(int),I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_2660_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xa)),archimedean_round(A,Ya)) ) ) ).

% round_mono
tff(fact_2661_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xa)),archimedean_round(A,Xa)) ) ).

% floor_le_round
tff(fact_2662_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xa)),archimedean_ceiling(A,Xa)) ) ).

% ceiling_ge_round
tff(fact_2663_set__encode__inf,axiom,
    ! [A4: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),A4)
     => ( aa(set(nat),nat,nat_set_encode,A4) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_2664_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D2)
     => ( ! [X3: int,K2: int] :
            ( aa(int,$o,P,X3)
          <=> aa(int,$o,P,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2))) )
       => ( ? [X_13: int] : aa(int,$o,P,X_13)
        <=> ? [X: int] :
              ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_2665_aset_I7_J,axiom,
    ! [D3: int,A4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb: int] :
                  ( member(int,Xb,A4)
                 => ( X2 != aa(int,int,minus_minus(int,Xb),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) ) ) ).

% aset(7)
tff(fact_2666_aset_I5_J,axiom,
    ! [D3: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,Ta,A4)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,A4)
                   => ( X2 != aa(int,int,minus_minus(int,Xb),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_2667_aset_I4_J,axiom,
    ! [D3: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,Ta,A4)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,A4)
                   => ( X2 != aa(int,int,minus_minus(int,Xb),Xa4) ) ) )
           => ( ( X2 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_2668_aset_I3_J,axiom,
    ! [D3: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A4)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,A4)
                   => ( X2 != aa(int,int,minus_minus(int,Xb),Xa4) ) ) )
           => ( ( X2 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_2669_bset_I7_J,axiom,
    ! [D3: int,Ta: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,Ta,B2)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,B2)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),X2)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ta),aa(int,int,minus_minus(int,X2),D3)) ) ) ) ) ).

% bset(7)
tff(fact_2670_bset_I5_J,axiom,
    ! [D3: int,B2: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb: int] :
                  ( member(int,Xb,B2)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X2),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,minus_minus(int,X2),D3)),Ta) ) ) ) ).

% bset(5)
tff(fact_2671_bset_I4_J,axiom,
    ! [D3: int,Ta: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,Ta,B2)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,B2)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa4) ) ) )
           => ( ( X2 != Ta )
             => ( aa(int,int,minus_minus(int,X2),D3) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_2672_bset_I3_J,axiom,
    ! [D3: int,Ta: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,aa(int,int,minus_minus(int,Ta),one_one(int)),B2)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,B2)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa4) ) ) )
           => ( ( X2 = Ta )
             => ( aa(int,int,minus_minus(int,X2),D3) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_2673_signed__take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% signed_take_bit_int_less_exp
tff(fact_2674_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(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,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),M)))) ) ).

% round_diff_minimal
tff(fact_2675_bset_I6_J,axiom,
    ! [D3: int,B2: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb: int] :
                  ( member(int,Xb,B2)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X2),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,X2),D3)),Ta) ) ) ) ).

% bset(6)
tff(fact_2676_bset_I8_J,axiom,
    ! [D3: int,Ta: int,B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,aa(int,int,minus_minus(int,Ta),one_one(int)),B2)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,B2)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X2)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,minus_minus(int,X2),D3)) ) ) ) ) ).

% bset(8)
tff(fact_2677_aset_I6_J,axiom,
    ! [D3: int,Ta: int,A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A4)
       => ! [X2: int] :
            ( ! [Xa4: int] :
                ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
               => ! [Xb: int] :
                    ( member(int,Xb,A4)
                   => ( X2 != aa(int,int,minus_minus(int,Xb),Xa4) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X2),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_2678_aset_I8_J,axiom,
    ! [D3: int,A4: set(int),Ta: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ! [X2: int] :
          ( ! [Xa4: int] :
              ( member(int,Xa4,set_or1337092689740270186AtMost(int,one_one(int),D3))
             => ! [Xb: int] :
                  ( member(int,Xb,A4)
                 => ( X2 != aa(int,int,minus_minus(int,Xb),Xa4) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D3)) ) ) ) ).

% aset(8)
tff(fact_2679_cpmi,axiom,
    ! [D3: int,P: fun(int,$o),P2: fun(int,$o),B2: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),X3),Z3)
           => ( aa(int,$o,P,X3)
            <=> aa(int,$o,P2,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,B2)
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
             => ( aa(int,$o,P,X3)
               => aa(int,$o,P,aa(int,int,minus_minus(int,X3),D3)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa(int,$o,P2,X3)
              <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
           => ( ? [X_13: int] : aa(int,$o,P,X_13)
            <=> ( ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P2,X) )
                | ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,B2)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_2680_cppi,axiom,
    ! [D3: int,P: fun(int,$o),P2: fun(int,$o),A4: set(int)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),D3)
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z3),X3)
           => ( aa(int,$o,P,X3)
            <=> aa(int,$o,P2,X3) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D3))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,A4)
                     => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
             => ( aa(int,$o,P,X3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D3)) ) )
         => ( ! [X3: int,K2: int] :
                ( aa(int,$o,P2,X3)
              <=> aa(int,$o,P2,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D3))) )
           => ( ? [X_13: int] : aa(int,$o,P,X_13)
            <=> ( ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & aa(int,$o,P2,X) )
                | ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D3))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,A4)
                        & aa(int,$o,P,aa(int,int,minus_minus(int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_2681_signed__take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_2682_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_2683_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2684_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2685_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2686_signed__take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),aa(int,int,minus_minus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_2687_signed__take__bit__int__eq__self,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2688_signed__take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2689_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,Na)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2690_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_2691_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Xa),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))) ) ).

% of_int_round_ge
tff(fact_2692_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,minus_minus(A,Xa),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))) ) ).

% of_int_round_gt
tff(fact_2693_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,Xa))),Xa))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_2694_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xa),aa(int,A,ring_1_of_int(A),Na)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2))))
         => ( archimedean_round(A,Xa) = Na ) ) ) ).

% round_unique'
tff(fact_2695_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xa: A] :
          archimedean_round(A,Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,Xa)),archimedean_ceiling(A,Xa),archim6421214686448440834_floor(A,Xa)) ) ).

% round_altdef
tff(fact_2696_tanh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),Xa)) = divide_divide(real,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2697_log__base__10__eq1,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,ln_ln(real),exp(real,one_one(real))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))))),aa(real,real,ln_ln(real),Xa)) ) ) ).

% log_base_10_eq1
tff(fact_2698_central__binomial__lower__bound,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),Na),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),Na)))),aa(nat,real,semiring_1_of_nat(real),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),Na))) ) ).

% central_binomial_lower_bound
tff(fact_2699_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A3,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2700_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = divide_divide(A,A3,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2701_log__base__10__eq2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),Xa) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(bit1(bit0(one2))))),exp(real,one_one(real)))),aa(real,real,ln_ln(real),Xa)) ) ) ).

% log_base_10_eq2
tff(fact_2702_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),one_one(nat))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_2703_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),zero_zero(A)) ) ).

% dvd_0_right
tff(fact_2704_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),A3)
        <=> ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_2705_dvd__1__left,axiom,
    ! [K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,suc,zero_zero(nat))),K) ).

% dvd_1_left
tff(fact_2706_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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_2707_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
    <=> ( ( K = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2708_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_2709_tanh__real__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xa)),aa(real,real,tanh(real),Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ).

% tanh_real_less_iff
tff(fact_2710_tanh__real__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tanh(real),Xa)),aa(real,real,tanh(real),Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ).

% tanh_real_le_iff
tff(fact_2711_semiring__norm_I73_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(73)
tff(fact_2712_semiring__norm_I80_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit1(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(80)
tff(fact_2713_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3))
        <=> ( ( C3 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_2714_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3))
        <=> ( ( C3 = zero_zero(A) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_2715_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),C3) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2716_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),C3) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2717_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3)
         => ( modulo_modulo(A,B3,A3) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_2718_binomial__1,axiom,
    ! [Na: nat] : binomial(Na,aa(nat,nat,suc,zero_zero(nat))) = Na ).

% binomial_1
tff(fact_2719_binomial__0__Suc,axiom,
    ! [K: nat] : binomial(zero_zero(nat),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_2720_binomial__eq__0__iff,axiom,
    ! [Na: nat,K: nat] :
      ( ( binomial(Na,K) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K) ) ).

% binomial_eq_0_iff
tff(fact_2721_binomial__n__0,axiom,
    ! [Na: nat] : binomial(Na,zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2722_semiring__norm_I72_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(72)
tff(fact_2723_semiring__norm_I81_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit1(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(81)
tff(fact_2724_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),one2) ).

% semiring_norm(70)
tff(fact_2725_semiring__norm_I77_J,axiom,
    ! [Na: num] : aa(num,$o,aa(num,fun(num,$o),ord_less(num),one2),bit1(Na)) ).

% semiring_norm(77)
tff(fact_2726_tanh__real__pos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tanh(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa) ) ).

% tanh_real_pos_iff
tff(fact_2727_tanh__real__neg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_2728_tanh__real__nonneg__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tanh(real),Xa))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% tanh_real_nonneg_iff
tff(fact_2729_tanh__real__nonpos__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tanh(real),Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_2730_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),bit1(K)) ) ).

% dbl_inc_simps(5)
tff(fact_2731_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Na: nat,A3: A,B3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2732_zero__less__binomial__iff,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),binomial(Na,K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ).

% zero_less_binomial_iff
tff(fact_2733_semiring__norm_I74_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit1(M)),bit0(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ).

% semiring_norm(74)
tff(fact_2734_semiring__norm_I79_J,axiom,
    ! [M: num,Na: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less(num),bit0(M)),bit1(Na))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ).

% semiring_norm(79)
tff(fact_2735_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),bit1(one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_2736_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W2)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2737_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2738_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,power_power(A,A3),Na)),zero_zero(A))
        <=> ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_2739_odd__Suc__minus__one,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))) = Na ) ) ).

% odd_Suc_minus_one
tff(fact_2740_even__diff__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Na))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ) ) ).

% even_diff_nat
tff(fact_2741_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),bit1(one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_2742_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W2)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W2) = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & ( A3 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2743_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,power_power(A,A3),Na))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% even_power
tff(fact_2744_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W2: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),aa(num,nat,numeral_numeral(nat),W2))),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W2))
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W2))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2745_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2746_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3)
        <=> ( ( A3 = zero_zero(A) )
           => ( B3 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_2747_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),A3)
         => ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_2748_gcd__nat_Oextremum,axiom,
    ! [A3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A3),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_2749_gcd__nat_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A3)
        & ( zero_zero(nat) != A3 ) ) ).

% gcd_nat.extremum_strict
tff(fact_2750_gcd__nat_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A3)
    <=> ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_2751_gcd__nat_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),A3),zero_zero(nat))
        & ( A3 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_2752_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),zero_zero(nat)),A3)
     => ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_2753_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% dvd_diff_nat
tff(fact_2754_binomial__eq__0,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K)
     => ( binomial(Na,K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2755_binomial__symmetric,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( binomial(Na,K) = binomial(Na,aa(nat,nat,minus_minus(nat,Na),K)) ) ) ).

% binomial_symmetric
tff(fact_2756_binomial__le__pow,axiom,
    ! [R2: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,R2)),aa(nat,nat,power_power(nat,Na),R2)) ) ).

% binomial_le_pow
tff(fact_2757_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),zero_zero(A)),one_one(A)) ) ).

% not_is_unit_0
tff(fact_2758_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),S3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),S3)) ) ) ) ).

% pinf(9)
tff(fact_2759_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z2),X2)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),S3))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),S3)) ) ) ) ).

% pinf(10)
tff(fact_2760_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),S3))
          <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),S3)) ) ) ) ).

% minf(9)
tff(fact_2761_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X2),Z2)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),S3))
          <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),S3)) ) ) ) ).

% minf(10)
tff(fact_2762_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),A3)
         => ( ( divide_divide(A,A3,B3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2763_num_Oexhaust,axiom,
    ! [Ya: num] :
      ( ( Ya != one2 )
     => ( ! [X24: num] : Ya != bit0(X24)
       => ~ ! [X32: num] : Ya != bit1(X32) ) ) ).

% num.exhaust
tff(fact_2764_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( ( modulo_modulo(A,A3,B3) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),A3) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2765_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3)
        <=> ( modulo_modulo(A,B3,A3) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2766_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] :
          ( ( modulo_modulo(A,A3,B3) = zero_zero(A) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),A3) ) ) ).

% mod_0_imp_dvd
tff(fact_2767_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A3),M)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% le_imp_power_dvd
tff(fact_2768_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat,B3: A,M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A3),Na)),B3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A3),M)),B3) ) ) ) ).

% power_le_dvd
tff(fact_2769_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xa: A,Ya: A,Na: nat,M: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xa),Ya)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,Xa),Na)),aa(nat,A,power_power(A,Ya),M)) ) ) ) ).

% dvd_power_le
tff(fact_2770_dvd__pos__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M) ) ) ).

% dvd_pos_nat
tff(fact_2771_nat__dvd__not__less,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
       => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M) ) ) ).

% nat_dvd_not_less
tff(fact_2772_dvd__minus__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),aa(nat,nat,minus_minus(nat,Na),M))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
        | aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na) ) ) ).

% dvd_minus_self
tff(fact_2773_zdvd__antisym__nonneg,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
       => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),M),Na)
         => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Na),M)
           => ( M = Na ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2774_dvd__diffD,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,minus_minus(nat,M),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Na)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),M) ) ) ) ).

% dvd_diffD
tff(fact_2775_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(nat,nat,minus_minus(nat,M),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),M)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Na) ) ) ) ).

% dvd_diffD1
tff(fact_2776_less__eq__dvd__minus,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_2777_zdvd__not__zless,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),M),Na)
       => ~ aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Na),M) ) ) ).

% zdvd_not_zless
tff(fact_2778_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),semiring_char_0_fact(A,Na)),semiring_char_0_fact(A,M)) ) ) ).

% fact_dvd
tff(fact_2779_dvd__Gcd__fin__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A),B3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4))
          <=> ! [X: A] :
                ( member(A,X,A4)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),X) ) ) ) ) ).

% dvd_Gcd_fin_iff
tff(fact_2780_Gcd__fin__greatest,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [B5: A] :
                ( member(A,B5,A4)
               => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B5) )
           => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4)) ) ) ) ).

% Gcd_fin_greatest
tff(fact_2781_tanh__real__lt__1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tanh(real),Xa)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_2782_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(int,nat,nat2,Z)),M)
    <=> $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),aa(nat,int,semiring_1_of_nat(int),M)),M = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_2783_zero__less__binomial,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),binomial(Na,K)) ) ).

% zero_less_binomial
tff(fact_2784_choose__mult,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),binomial(Na,M)),binomial(M,K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),binomial(Na,K)),binomial(aa(nat,nat,minus_minus(nat,Na),K),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_2785_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [C5: A] : B3 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C5) ) ) ) ).

% unit_dvdE
tff(fact_2786_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X))
        <=> ? [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),zero_zero(A)))
              & aa(A,$o,P,X) ) ) ) ).

% unity_coeff_ex
tff(fact_2787_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C3: A,B3: A,D2: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( C3 != zero_zero(A) )
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3)
             => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),D2)
               => ( ( divide_divide(A,B3,A3) = divide_divide(A,D2,C3) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),C3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2788_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C3: A,B3: A,A3: A] :
          ( ( C3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),C3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),divide_divide(A,B3,C3))
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3)),B3) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2789_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),divide_divide(A,A3,B3)),C3)
            <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C3),B3)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2790_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3)
           => ( ( divide_divide(A,B3,A3) = C3 )
            <=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2791_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),one_one(A))
         => ( ( divide_divide(A,A3,B3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2792_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bit1(Na)) = 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),Na)),aa(num,A,numeral_numeral(A),Na))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_2793_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),one_one(A))
         => ( modulo_modulo(A,A3,B3) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_2794_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,A3),Na)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
            | ( Na = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_2795_eval__nat__numeral_I3_J,axiom,
    ! [Na: num] : aa(num,nat,numeral_numeral(nat),bit1(Na)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(Na))) ).

% eval_nat_numeral(3)
tff(fact_2796_dvd__imp__le,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na) ) ) ).

% dvd_imp_le
tff(fact_2797_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2798_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),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),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na) ) ) ).

% dvd_mult_cancel
tff(fact_2799_bezout__add__strong__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [D6: nat,X3: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D6),A3)
          & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),D6),B3)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y)),D6) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2800_zdvd__imp__le,axiom,
    ! [Z: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),Z),Na)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),Na) ) ) ).

% zdvd_imp_le
tff(fact_2801_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,Na))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2802_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),I)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),I)) ) ) ).

% dvd_imp_le_int
tff(fact_2803_mod__eq__dvd__iff__nat,axiom,
    ! [Na: nat,M: nat,Q5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( ( modulo_modulo(nat,M,Q5) = modulo_modulo(nat,Na,Q5) )
      <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Q5),aa(nat,nat,minus_minus(nat,M),Na)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2804_dvd__fact,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),semiring_char_0_fact(nat,Na)) ) ) ).

% dvd_fact
tff(fact_2805_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(int,nat,nat2,K))
      <=> aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_nat_iff
tff(fact_2806_tanh__real__gt__neg1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),Xa)) ).

% tanh_real_gt_neg1
tff(fact_2807_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A)) ) ).

% even_zero
tff(fact_2808_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),one_one(A))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)) = divide_divide(A,one_one(A),B3) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2809_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),one_one(A))
           => ( divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = divide_divide(A,one_one(A),B3) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2810_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B5: A] :
                  ( ( B5 != zero_zero(A) )
                 => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B5),one_one(A))
                   => ( ( divide_divide(A,one_one(A),A3) = B5 )
                     => ( ( divide_divide(A,one_one(A),B5) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B5) = one_one(A) )
                         => ( divide_divide(A,C3,A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C3),B5) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2811_odd__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
     => ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_2812_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xa: A,M: nat,Na: nat] :
          ( ( Xa != zero_zero(A) )
         => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,Xa),M)),aa(nat,A,power_power(A,Xa),Na))
          <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xa),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ) ).

% dvd_power_iff
tff(fact_2813_binomial__fact__lemma,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( 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,minus_minus(nat,Na),K)))),binomial(Na,K)) = semiring_char_0_fact(nat,Na) ) ) ).

% binomial_fact_lemma
tff(fact_2814_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num,Q5: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),bit1(Na)),aa(num,A,numeral_numeral(A),bit0(Q5))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2815_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat,Xa: A] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
            | ( Xa = one_one(A) ) )
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Xa),aa(nat,A,power_power(A,Xa),Na)) ) ) ).

% dvd_power
tff(fact_2816_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),bit1(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_2817_Suc3__eq__add__3,axiom,
    ! [Na: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Na))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit1(one2))),Na) ).

% Suc3_eq_add_3
tff(fact_2818_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K)))),semiring_char_0_fact(A,Na)) ) ) ).

% choose_dvd
tff(fact_2819_dvd__mult__cancel1,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),M)
      <=> ( Na = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2820_dvd__mult__cancel2,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),M)),M)
      <=> ( Na = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2821_dvd__minus__add,axiom,
    ! [Q5: nat,Na: nat,R2: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q5),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q5),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),aa(nat,nat,minus_minus(nat,Na),Q5))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M)),Q5))) ) ) ) ).

% dvd_minus_add
tff(fact_2822_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,power_power(nat,I),M)),aa(nat,nat,power_power(nat,I),Na))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% power_dvd_imp_le
tff(fact_2823_mod__nat__eqI,axiom,
    ! [R2: nat,Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),R2),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R2),M)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),aa(nat,nat,minus_minus(nat,M),R2))
         => ( modulo_modulo(nat,M,Na) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2824_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),L),K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) )
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_2825_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Na),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),binomial(Na,K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2826_binomial__maximum_H,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),K)),binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na),Na)) ).

% binomial_maximum'
tff(fact_2827_binomial__mono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,K)),binomial(Na,K7)) ) ) ).

% binomial_mono
tff(fact_2828_binomial__maximum,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,K)),binomial(Na,divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_2829_binomial__antimono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,K7)),binomial(Na,K)) ) ) ) ).

% binomial_antimono
tff(fact_2830_binomial__le__pow2,axiom,
    ! [Na: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),binomial(Na,K)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% binomial_le_pow2
tff(fact_2831_choose__reduce__nat,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => ( binomial(Na,K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat)),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat)),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2832_times__binomial__minus1__eq,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),binomial(Na,K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat)),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2833_num_Osize_I6_J,axiom,
    ! [X33: num] : aa(num,nat,size_size(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_2834_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
        <=> ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2835_binomial__altdef__nat,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
     => ( binomial(Na,K) = divide_divide(nat,semiring_char_0_fact(nat,Na),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,minus_minus(nat,Na),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_2836_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q5: num,Na: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q5))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),bit1(Na)),aa(num,A,numeral_numeral(A),bit0(Q5))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),Na),aa(num,A,numeral_numeral(A),Q5)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2837_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q5: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),bit1(M)),aa(num,A,numeral_numeral(A),bit0(Q5))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),bit0(Q5))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q5)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2838_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A,B3: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ).

% power_mono_odd
tff(fact_2839_odd__pos,axiom,
    ! [Na: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% odd_pos
tff(fact_2840_card__3__iff,axiom,
    ! [A: $tType,S: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S) = aa(num,nat,numeral_numeral(nat),bit1(one2)) )
    <=> ? [X: A,Y4: A,Z5: A] :
          ( ( S = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z5),bot_bot(set(A))))) )
          & ( X != Y4 )
          & ( Y4 != Z5 )
          & ( X != Z5 ) ) ) ).

% card_3_iff
tff(fact_2841_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),A4))
     => ( A4 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_2842_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(nat,nat,power_power(nat,K),M)),aa(nat,nat,power_power(nat,K),Na))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% dvd_power_iff_le
tff(fact_2843_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,M,A3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2844_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,M,A3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_2845_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),aa(num,real,numeral_numeral(real),bit1(one2))) ).

% exp_le
tff(fact_2846_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),binomial(Na,K)),binomial(Na,aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_2847_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K7),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),binomial(Na,K7)),binomial(Na,K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_2848_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),K7)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K7)),Na)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),binomial(Na,K)),binomial(Na,K7)) ) ) ).

% binomial_strict_mono
tff(fact_2849_binomial__addition__formula,axiom,
    ! [Na: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( binomial(Na,aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat)),aa(nat,nat,suc,K))),binomial(aa(nat,nat,minus_minus(nat,Na),one_one(nat)),K)) ) ) ).

% binomial_addition_formula
tff(fact_2850_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( 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),binomial(Na,K))) = divide_divide(A,semiring_char_0_fact(A,Na),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K))) ) ) ) ).

% fact_binomial
tff(fact_2851_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),binomial(Na,K)) = divide_divide(A,semiring_char_0_fact(A,Na),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,minus_minus(nat,Na),K)))) ) ) ) ).

% binomial_fact
tff(fact_2852_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
           => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
             => ( modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_2853_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_2854_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),Na))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2855_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A] :
          ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),Na))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3) ) ) ) ).

% zero_le_odd_power
tff(fact_2856_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A3),Na)) ) ) ).

% zero_le_even_power
tff(fact_2857_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: A,B3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A3)),aa(A,A,abs_abs(A),B3))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),Na)),aa(nat,A,power_power(A,B3),Na)) ) ) ) ).

% power_mono_even
tff(fact_2858_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit1(one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_2859_even__set__encode__iff,axiom,
    ! [A4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A4))
      <=> ~ member(nat,zero_zero(nat),A4) ) ) ).

% even_set_encode_iff
tff(fact_2860_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A3),Na))
        <=> ( ( Na = zero_zero(nat) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & ( A3 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2861_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ).

% even_mask_div_iff'
tff(fact_2862_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,power_power(A,A3),Na)),zero_zero(A))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
            & ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),zero_zero(A)) )
              | ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2863_even__mod__4__div__2,axiom,
    ! [Na: nat] :
      ( ( modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_2864_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% even_mask_div_iff
tff(fact_2865_Bernoulli__inequality__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),Xa))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xa)),Na)) ) ).

% Bernoulli_inequality_even
tff(fact_2866_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)))
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            | ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
              & aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2867_sin__coeff__def,axiom,
    ! [X2: nat] :
      sin_coeff(X2) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),X2),zero_zero(real),divide_divide(real,aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,aa(nat,nat,minus_minus(nat,X2),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),semiring_char_0_fact(real,X2))) ).

% sin_coeff_def
tff(fact_2868_binomial__code,axiom,
    ! [Na: nat,K: nat] :
      binomial(Na,K) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K),
        zero_zero(nat),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),binomial(Na,aa(nat,nat,minus_minus(nat,Na),K)),divide_divide(nat,set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,Na),K)),one_one(nat)),Na,one_one(nat)),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_2869_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : size_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_2870_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3) = $ite(Na = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_2871_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : size_num(bit0(X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_2872_set__decode__plus__power__2,axiom,
    ! [Na: nat,Z: nat] :
      ( ~ member(nat,Na,nat_set_decode(Z))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),Z)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Na),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_2873_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_2874_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A3) = zero_zero(A) ) ).

% take_bit_0
tff(fact_2875_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_2876_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_2877_set__encode__inverse,axiom,
    ! [A4: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),A4)
     => ( nat_set_decode(aa(set(nat),nat,nat_set_encode,A4)) = A4 ) ) ).

% set_encode_inverse
tff(fact_2878_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),one_one(A)) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_2879_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3))
        <=> ( ( Na = zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% even_take_bit_eq
tff(fact_2880_set__decode__0,axiom,
    ! [Xa: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xa))
    <=> ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xa) ) ).

% set_decode_0
tff(fact_2881_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A3) = modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_2882_dvd__antisym,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),M),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M)
       => ( M = Na ) ) ) ).

% dvd_antisym
tff(fact_2883_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A,B3: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),B3) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B3) ) ) ) ) ).

% take_bit_tightened
tff(fact_2884_take__bit__nat__less__eq__self,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_2885_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Na: nat,Q5: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q5)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),Q5)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_2886_nat__take__bit__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(int,nat,nat2,K)) ) ) ).

% nat_take_bit_eq
tff(fact_2887_take__bit__nat__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(int,nat,nat2,K)) = aa(int,nat,nat2,aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ) ).

% take_bit_nat_eq
tff(fact_2888_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Na: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_2889_take__bit__int__less__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_2890_take__bit__nonnegative,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ).

% take_bit_nonnegative
tff(fact_2891_not__take__bit__negative,axiom,
    ! [Na: nat,K: int] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_2892_take__bit__int__greater__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_2893_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A3: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),bit_se2584673776208193580ke_bit(A,Na),bit_ri4674362597316999326ke_bit(A,M)),
            A3) ) ).

% signed_take_bit_take_bit
tff(fact_2894_finite__set__decode,axiom,
    ! [Na: nat] : aa(set(nat),$o,finite_finite2(nat),nat_set_decode(Na)) ).

% finite_set_decode
tff(fact_2895_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),bit_se2638667681897837118et_bit(A,M,A3)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3),bit_se2638667681897837118et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3))) ) ).

% take_bit_unset_bit_eq
tff(fact_2896_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),bit_se5668285175392031749et_bit(A,M,A3)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3),bit_se5668285175392031749et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3))) ) ).

% take_bit_set_bit_eq
tff(fact_2897_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,Na),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_2898_subset__decode__imp__le,axiom,
    ! [M: nat,Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% subset_decode_imp_le
tff(fact_2899_take__bit__nat__eq__self,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_2900_take__bit__nat__less__exp,axiom,
    ! [Na: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% take_bit_nat_less_exp
tff(fact_2901_take__bit__nat__eq__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M) = M )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2902_take__bit__int__less__exp,axiom,
    ! [Na: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ).

% take_bit_int_less_exp
tff(fact_2903_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb3: nat,Xc: A,Ya: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,Xb3,Xc) = Ya )
     => ( Ya = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb3),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb3,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2904_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B3: nat,Acc: A] :
      set_fo6178422350223883121st_nat(A,F2,A3,B3,Acc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),A3),Acc,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B3,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2905_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_2906_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3) = zero_zero(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),A3) ) ) ).

% take_bit_eq_0_iff
tff(fact_2907_take__bit__nat__less__self__iff,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),M)),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_2908_take__bit__int__less__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_2909_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2910_take__bit__int__eq__self,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2911_take__bit__int__eq__self__iff,axiom,
    ! [Na: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2912_take__bit__int__less__eq,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)),aa(int,int,minus_minus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))) ) ) ).

% take_bit_int_less_eq
tff(fact_2913_take__bit__int__greater__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))),aa(int,int,bit_se2584673776208193580ke_bit(int,Na),K)) ) ).

% take_bit_int_greater_eq
tff(fact_2914_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( ( divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3) = $ite(aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3),zero_zero(A),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2915_take__bit__minus__small__eq,axiom,
    ! [K: int,Na: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Na),aa(int,int,uminus_uminus(int),K)) = aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2916_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Na))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,Na))),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,Na))))) ) ).

% modulo_int_unfold
tff(fact_2917_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Na: nat] :
      divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),Na))) = $ite(
        ( ( sgn_sgn(int,L) = zero_zero(int) )
        | ( sgn_sgn(int,K) = zero_zero(int) )
        | ( Na = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(sgn_sgn(int,K) = sgn_sgn(int,L),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,M,Na)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,M,Na)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Na),M)))))) ) ).

% divide_int_unfold
tff(fact_2918_sqrt__sum__squares__half__less,axiom,
    ! [Xa: real,U: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,U,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),divide_divide(real,U,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_2919_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,M,A3))
        <=> ~ ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_2920_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : modulo_modulo(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% one_mod_2_pow_eq
tff(fact_2921_flip__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Na,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2922_flip__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se8732182000553998342ip_bit(int,Na,K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_2923_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_2924_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$false) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_2925_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = zero_zero(A) )
        <=> ~ (P) ) ) ).

% of_bool_eq_0_iff
tff(fact_2926_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( ~ (P)
            & (Q) ) ) ) ).

% of_bool_less_iff
tff(fact_2927_real__sqrt__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ).

% real_sqrt_less_iff
tff(fact_2928_real__sqrt__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ).

% real_sqrt_le_iff
tff(fact_2929_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P)))
        <=> (P) ) ) ).

% zero_less_of_bool_iff
tff(fact_2930_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A))
        <=> ~ (P) ) ) ).

% of_bool_less_one_iff
tff(fact_2931_real__sqrt__gt__0__iff,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya) ) ).

% real_sqrt_gt_0_iff
tff(fact_2932_real__sqrt__lt__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_2933_real__sqrt__le__0__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_2934_real__sqrt__ge__0__iff,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya) ) ).

% real_sqrt_ge_0_iff
tff(fact_2935_real__sqrt__lt__1__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_2936_real__sqrt__gt__1__iff,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(real,real,sqrt,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Ya) ) ).

% real_sqrt_gt_1_iff
tff(fact_2937_real__sqrt__le__1__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_2938_real__sqrt__ge__1__iff,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Ya))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Ya) ) ).

% real_sqrt_ge_1_iff
tff(fact_2939_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o] : aa(nat,A,semiring_1_of_nat(A),aa($o,nat,zero_neq_one_of_bool(nat),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% of_nat_of_bool
tff(fact_2940_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A3)),sgn_sgn(A,A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_2941_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,abs_abs(A),sgn_sgn(A,A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_2942_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : sgn_sgn(A,aa(A,A,abs_abs(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_2943_Suc__0__mod__eq,axiom,
    ! [Na: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa($o,nat,zero_neq_one_of_bool(nat),Na != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_2944_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% take_bit_of_1
tff(fact_2945_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat] : sgn_sgn(A,aa(nat,A,semiring_1_of_nat(A),Na)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ) ).

% sgn_of_nat
tff(fact_2946_take__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)) ).

% take_bit_of_Suc_0
tff(fact_2947_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B3: $o] : divide_divide(A,aa($o,A,zero_neq_one_of_bool(A),(B3)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_2948_real__sqrt__pow2__iff,axiom,
    ! [Xa: real] :
      ( ( aa(nat,real,power_power(real,aa(real,real,sqrt,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa) ) ).

% real_sqrt_pow2_iff
tff(fact_2949_real__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(nat,real,power_power(real,aa(real,real,sqrt,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa ) ) ).

% real_sqrt_pow2
tff(fact_2950_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] : divide_divide(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_2951_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: nat] : divide_divide(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_2952_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) ) ).

% take_bit_of_exp
tff(fact_2953_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_2954_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_2955_real__sqrt__less__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Ya)) ) ).

% real_sqrt_less_mono
tff(fact_2956_real__sqrt__le__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Ya)) ) ).

% real_sqrt_le_mono
tff(fact_2957_real__sqrt__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_gt_zero
tff(fact_2958_real__sqrt__eq__zero__cancel,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( aa(real,real,sqrt,Xa) = zero_zero(real) )
       => ( Xa = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_2959_real__sqrt__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_ge_zero
tff(fact_2960_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_2961_real__sqrt__ge__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Xa)) ) ).

% real_sqrt_ge_one
tff(fact_2962_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_2963_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P3: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P3)) = $ite((P3),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_2964_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ( ( (P3)
             => aa(A,$o,P,one_one(A)) )
            & ( ~ (P3)
             => aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_2965_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P3: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P3)))
        <=> ~ ( ( (P3)
                & ~ aa(A,$o,P,one_one(A)) )
              | ( ~ (P3)
                & ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_2966_real__div__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( divide_divide(real,Xa,aa(real,real,sqrt,Xa)) = aa(real,real,sqrt,Xa) ) ) ).

% real_div_sqrt
tff(fact_2967_sqrt__add__le__add__sqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Ya))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,Xa)),aa(real,real,sqrt,Ya))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_2968_le__real__sqrt__sumsq,axiom,
    ! [Xa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),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),Xa),Xa)),aa(real,real,aa(real,fun(real,real),times_times(real),Ya),Ya)))) ).

% le_real_sqrt_sumsq
tff(fact_2969_sqrt2__less__2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sqrt2_less_2
tff(fact_2970_sqrt__divide__self__eq,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( divide_divide(real,aa(real,real,sqrt,Xa),Xa) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa)) ) ) ).

% sqrt_divide_self_eq
tff(fact_2971_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,M: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Na),bit_se8732182000553998342ip_bit(A,M,A3)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3),bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3))) ) ).

% take_bit_flip_bit_eq
tff(fact_2972_real__less__rsqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Ya)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,sqrt,Ya)) ) ).

% real_less_rsqrt
tff(fact_2973_real__le__rsqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Ya)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sqrt,Ya)) ) ).

% real_le_rsqrt
tff(fact_2974_sqrt__le__D,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),Ya)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sqrt_le_D
tff(fact_2975_real__sqrt__unique,axiom,
    ! [Ya: real,Xa: real] :
      ( ( aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xa )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( aa(real,real,sqrt,Xa) = Ya ) ) ) ).

% real_sqrt_unique
tff(fact_2976_real__le__lsqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xa)),Ya) ) ) ) ).

% real_le_lsqrt
tff(fact_2977_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),U)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_2978_real__sqrt__sum__squares__ge1,axiom,
    ! [Xa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_2979_real__sqrt__sum__squares__ge2,axiom,
    ! [Ya: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_2980_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A3: real,C3: real,B3: real,D2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),C3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),B3),D2)),aa(num,nat,numeral_numeral(nat),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,power_power(real,A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,C3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,D2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_2981_sqrt__ge__absD,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,sqrt,Ya))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Ya) ) ).

% sqrt_ge_absD
tff(fact_2982_real__less__lsqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,Xa)),Ya) ) ) ) ).

% real_less_lsqrt
tff(fact_2983_sqrt__sum__squares__le__sum,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Ya)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_2984_real__inv__sqrt__pow2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(nat,real,power_power(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),Xa) ) ) ).

% real_inv_sqrt_pow2
tff(fact_2985_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,abs_abs(real),Ya))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_2986_real__sqrt__ge__abs2,axiom,
    ! [Ya: real,Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs2
tff(fact_2987_real__sqrt__ge__abs1,axiom,
    ! [Xa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs1
tff(fact_2988_ln__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,Xa)) = divide_divide(real,aa(real,real,ln_ln(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_2989_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Na: nat] : modulo_modulo(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).

% exp_mod_exp
tff(fact_2990_arsinh__real__aux,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_2991_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xa: real,Ya: real,Xaa: real,Yaa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Yaa),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_2992_real__sqrt__power__even,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
       => ( aa(nat,real,power_power(real,aa(real,real,sqrt,Xa)),Na) = aa(nat,real,power_power(real,Xa),divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_2993_arith__geo__mean__sqrt,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xa),Ya))),divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),Ya),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_2994_powr__half__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( powr(real,Xa,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,sqrt,Xa) ) ) ).

% powr_half_sqrt
tff(fact_2995_cos__x__y__le__one,axiom,
    ! [Xa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),divide_divide(real,Xa,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),one_one(real)) ).

% cos_x_y_le_one
tff(fact_2996_real__sqrt__sum__squares__less,axiom,
    ! [Xa: real,U: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Ya)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_2997_arcosh__real__def,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xa)
     => ( aa(real,real,arcosh(real),Xa) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_2998_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          divide_divide(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) )
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M) ))),
            aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Na))) ) ).

% exp_div_exp_eq
tff(fact_2999_cosh__ln__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( cosh(real,aa(real,real,ln_ln(real),Xa)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),aa(real,real,inverse_inverse(real),Xa)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_3000_Suc__0__xor__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% Suc_0_xor_eq
tff(fact_3001_xor__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% xor_Suc_0_eq
tff(fact_3002_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cosh(A,Xa) = zero_zero(A) )
        <=> ( aa(nat,A,power_power(A,exp(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_3003_xor__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Na) = $ite(
        M = zero_zero(nat),
        Na,
        $ite(Na = zero_zero(nat),M,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),bit0(one2)))),modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_3004_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),zero_zero(A)) = A3 ) ).

% xor.right_neutral
tff(fact_3005_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A3) = A3 ) ).

% xor.left_neutral
tff(fact_3006_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),A3) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_3007_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xa),Xa) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_3008_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_3009_xor__nat__numerals_I1_J,axiom,
    ! [Ya: 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),bit0(Ya))) = aa(num,nat,numeral_numeral(nat),bit1(Ya)) ).

% xor_nat_numerals(1)
tff(fact_3010_xor__nat__numerals_I2_J,axiom,
    ! [Ya: 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),bit1(Ya))) = aa(num,nat,numeral_numeral(nat),bit0(Ya)) ).

% xor_nat_numerals(2)
tff(fact_3011_xor__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xa)) ).

% xor_nat_numerals(3)
tff(fact_3012_xor__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit1(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(Xa)) ).

% xor_nat_numerals(4)
tff(fact_3013_cosh__real__pos,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cosh(real,Xa)) ).

% cosh_real_pos
tff(fact_3014_cosh__real__nonpos__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xa)),cosh(real,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),Xa) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_3015_cosh__real__nonneg__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xa)),cosh(real,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_3016_cosh__real__nonneg,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cosh(real,Xa)) ).

% cosh_real_nonneg
tff(fact_3017_arcosh__cosh__real,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,real,arcosh(real),cosh(real,Xa)) = Xa ) ) ).

% arcosh_cosh_real
tff(fact_3018_cosh__real__ge__1,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),cosh(real,Xa)) ).

% cosh_real_ge_1
tff(fact_3019_sinh__less__cosh__real,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),sinh(real,Xa)),cosh(real,Xa)) ).

% sinh_less_cosh_real
tff(fact_3020_sinh__le__cosh__real,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xa)),cosh(real,Xa)) ).

% sinh_le_cosh_real
tff(fact_3021_cosh__real__nonpos__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),Xa) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_3022_cosh__real__nonneg__less__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_3023_cosh__real__strict__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cosh(real,Xa)),cosh(real,Ya)) ) ) ).

% cosh_real_strict_mono
tff(fact_3024_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Ya: A] :
          ( ( cosh(A,Xa) != zero_zero(A) )
         => ( ( cosh(A,Ya) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),Xa)),aa(A,A,tanh(A),Ya)),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),Xa)),aa(A,A,tanh(A),Ya)))) ) ) ) ) ).

% tanh_add
tff(fact_3025_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Na),A3) = $ite(Na = zero_zero(nat),A3,aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat))),divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_3026_Suc__0__or__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% Suc_0_or_eq
tff(fact_3027_or__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) ).

% or_Suc_0_eq
tff(fact_3028_and__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Na) = $ite(
        ( ( M = zero_zero(nat) )
        | ( Na = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_3029_is__empty__set,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( is_empty(A,aa(list(A),set(A),set2(A),Xs))
    <=> null(A,Xs) ) ).

% is_empty_set
tff(fact_3030_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_3031_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_3032_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xa) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_3033_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_3034_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A3) = A3 ) ).

% or.left_neutral
tff(fact_3035_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),zero_zero(A)) = A3 ) ).

% or.right_neutral
tff(fact_3036_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_3037_drop__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4197421643247451524op_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_3038_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_3039_drop__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4197421643247451524op_bit(int,Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_3040_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_3041_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,B3: $o] :
          aa(A,A,bit_se4197421643247451524op_bit(A,Na),aa($o,A,zero_neq_one_of_bool(A),(B3))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( Na = zero_zero(nat) )
            & (B3) )) ) ).

% drop_bit_of_bool
tff(fact_3042_drop__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,Na),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),Na = zero_zero(nat)) ).

% drop_bit_of_Suc_0
tff(fact_3043_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,Na),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Na = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_3044_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Ya: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Ya))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_3045_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xa))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_3046_and__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_3047_and__nat__numerals_I1_J,axiom,
    ! [Ya: 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),bit0(Ya))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_3048_or__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit1(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xa)) ).

% or_nat_numerals(4)
tff(fact_3049_or__nat__numerals_I2_J,axiom,
    ! [Ya: 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),bit1(Ya))) = aa(num,nat,numeral_numeral(nat),bit1(Ya)) ).

% or_nat_numerals(2)
tff(fact_3050_or__nat__numerals_I3_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(Xa))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit1(Xa)) ).

% or_nat_numerals(3)
tff(fact_3051_or__nat__numerals_I1_J,axiom,
    ! [Ya: 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),bit0(Ya))) = aa(num,nat,numeral_numeral(nat),bit1(Ya)) ).

% or_nat_numerals(1)
tff(fact_3052_and__nat__numerals_I4_J,axiom,
    ! [Xa: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit1(Xa))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_3053_and__nat__numerals_I2_J,axiom,
    ! [Ya: 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),bit1(Ya))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_3054_Suc__0__and__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),Na) = modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_3055_and__Suc__0__eq,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Na),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_3056_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B3 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_3057_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),zero_zero(A)) = Xa ) ).

% bit.disj_zero_right
tff(fact_3058_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Xa) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Xa) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Ya) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Ya) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( Xa = Ya ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_3059_XOR__lower,axiom,
    ! [Xa: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Ya)) ) ) ).

% XOR_lower
tff(fact_3060_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3) = A3 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,Na),A3) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_3061_XOR__upper,axiom,
    ! [Xa: int,Na: nat,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ya),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xa),Ya)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% XOR_upper
tff(fact_3062_or__nat__unfold,axiom,
    ! [M: nat,Na: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Na) = $ite(
        M = zero_zero(nat),
        Na,
        $ite(Na = zero_zero(nat),M,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),bit0(one2)))),modulo_modulo(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),bit0(one2)))),divide_divide(nat,Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_3063_arctan__lbound,axiom,
    ! [Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Ya)) ).

% arctan_lbound
tff(fact_3064_arctan__bounded,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Ya))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Ya)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% arctan_bounded
tff(fact_3065_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Na)
        <=> $ite(Na = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3),aa(nat,$o,bit_se5641148757651400278ts_bit(A,divide_divide(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_3066_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),Na)
          <=> $ite(Na = zero_zero(nat),~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3),aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)),Na)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_3067_max_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),A3) = A3 ) ).

% max.idem
tff(fact_3068_max_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ).

% max.left_idem
tff(fact_3069_max_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ).

% max.right_idem
tff(fact_3070_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb1
tff(fact_3071_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb2
tff(fact_3072_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C3)),A3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3) ) ) ) ).

% max.bounded_iff
tff(fact_3073_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_3074_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb3
tff(fact_3075_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb4
tff(fact_3076_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_3077_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_3078_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_3079_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        | aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_3080_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int))
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_3081_max__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),bot_bot(A)),Xa) = Xa ) ).

% max_bot
tff(fact_3082_max__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),bot_bot(A)) = Xa ) ).

% max_bot2
tff(fact_3083_max__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)) ).

% max_Suc_Suc
tff(fact_3084_max__0R,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),zero_zero(nat)) = Na ).

% max_0R
tff(fact_3085_max__0L,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),Na) = Na ).

% max_0L
tff(fact_3086_max__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),zero_zero(nat)) = A3 ).

% max_nat.right_neutral
tff(fact_3087_max__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B3) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_3088_max__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),A3) = A3 ).

% max_nat.left_neutral
tff(fact_3089_max__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),A3),B3) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_3090_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),V2),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_3091_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xa)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(3)
tff(fact_3092_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xa)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(4)
tff(fact_3093_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_3094_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_3095_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),Xa)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(5)
tff(fact_3096_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),Xa)),one_one(A)) = aa(num,A,numeral_numeral(A),Xa) ) ).

% max_0_1(6)
tff(fact_3097_signed__take__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K))
    <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_3098_signed__take__bit__negative__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Na),K)),zero_zero(int))
    <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) ) ).

% signed_take_bit_negative_iff
tff(fact_3099_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_3100_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),V2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_3101_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_3102_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),zero_zero(nat))
        <=> ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% bit_0
tff(fact_3103_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A3,aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( ( Na = zero_zero(nat) )
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% bit_mod_2_iff
tff(fact_3104_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3),B3,A3) ) ).

% max_def
tff(fact_3105_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya) = Xa ) ) ) ).

% max_absorb1
tff(fact_3106_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya) = Ya ) ) ) ).

% max_absorb2
tff(fact_3107_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.coboundedI2
tff(fact_3108_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.coboundedI1
tff(fact_3109_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = B3 ) ) ) ).

% max.absorb_iff2
tff(fact_3110_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = A3 ) ) ) ).

% max.absorb_iff1
tff(fact_3111_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Ya) ) ) ) ).

% le_max_iff_disj
tff(fact_3112_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ).

% max.cobounded2
tff(fact_3113_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ).

% max.cobounded1
tff(fact_3114_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ) ) ).

% max.order_iff
tff(fact_3115_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C3)),A3) ) ) ) ).

% max.boundedI
tff(fact_3116_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C3)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3) ) ) ) ).

% max.boundedE
tff(fact_3117_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% max.orderI
tff(fact_3118_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) ) ) ) ).

% max.orderE
tff(fact_3119_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A3: A,D2: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ) ).

% max.mono
tff(fact_3120_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),B3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.strict_coboundedI2
tff(fact_3121_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)) ) ) ).

% max.strict_coboundedI1
tff(fact_3122_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
tff(fact_3123_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C3)),A3)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),A3) ) ) ) ).

% max.strict_boundedE
tff(fact_3124_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Ya) ) ) ) ).

% less_max_iff_disj
tff(fact_3125_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: nat,Ya: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,semiring_1_of_nat(A),Ya)) ) ).

% of_nat_max
tff(fact_3126_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_3127_max_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C3)) ) ).

% max.assoc
tff(fact_3128_max_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3) = aa(A,A,aa(A,fun(A,A),ord_max(A),B3),A3) ) ).

% max.commute
tff(fact_3129_max_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),B3),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C3)) ) ).

% max.left_commute
tff(fact_3130_sup__max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( sup_sup(A) = ord_max(A) ) ) ).

% sup_max
tff(fact_3131_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,minus_minus(A,Xa),Z)),aa(A,A,minus_minus(A,Ya),Z)) ) ).

% max_diff_distrib_left
tff(fact_3132_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),ord_max(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)) ) ).

% max_add_distrib_right
tff(fact_3133_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Ya: 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),Xa),Ya)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ya),Z)) ) ).

% max_add_distrib_left
tff(fact_3134_nat__mult__max__left,axiom,
    ! [M: nat,Na: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)) ).

% nat_mult_max_left
tff(fact_3135_nat__mult__max__right,axiom,
    ! [M: nat,Na: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)) ).

% nat_mult_max_right
tff(fact_3136_nat__add__max__left,axiom,
    ! [M: nat,Na: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),Na)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Q5)) ).

% nat_add_max_left
tff(fact_3137_nat__add__max__right,axiom,
    ! [M: nat,Na: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q5)) ).

% nat_add_max_right
tff(fact_3138_bit__nat__iff,axiom,
    ! [K: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(int,nat,nat2,K)),Na)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) ) ) ).

% bit_nat_iff
tff(fact_3139_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,one_one(A)),Na)
        <=> ( Na = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_3140_not__bit__Suc__0__Suc,axiom,
    ! [Na: nat] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,Na)) ).

% not_bit_Suc_0_Suc
tff(fact_3141_bit__Suc__0__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),Na)
    <=> ( Na = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_3142_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Na) ) ) ) ).

% bit_take_bit_iff
tff(fact_3143_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_3144_OR__lower,axiom,
    ! [Xa: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Ya)) ) ) ).

% OR_lower
tff(fact_3145_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B3: $o,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa($o,A,zero_neq_one_of_bool(A),(B3))),Na)
        <=> ( (B3)
            & ( Na = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_3146_AND__upper2_H,axiom,
    ! [Ya: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ya),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Ya)),Z) ) ) ).

% AND_upper2'
tff(fact_3147_AND__upper1_H,axiom,
    ! [Ya: int,Z: int,Yaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ya),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Ya),Yaa)),Z) ) ) ).

% AND_upper1'
tff(fact_3148_AND__upper2,axiom,
    ! [Ya: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Ya)),Ya) ) ).

% AND_upper2
tff(fact_3149_AND__upper1,axiom,
    ! [Xa: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Ya)),Xa) ) ).

% AND_upper1
tff(fact_3150_AND__lower,axiom,
    ! [Xa: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Ya)) ) ).

% AND_lower
tff(fact_3151_nat__minus__add__max,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,Na),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),M) ).

% nat_minus_add_max
tff(fact_3152_pi__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_3153_pi__not__less__zero,axiom,
    ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_3154_pi__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_3155_not__bit__Suc__0__numeral,axiom,
    ! [Na: num] : ~ aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),Na)) ).

% not_bit_Suc_0_numeral
tff(fact_3156_AND__upper2_H_H,axiom,
    ! [Ya: int,Z: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ya),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xa),Ya)),Z) ) ) ).

% AND_upper2''
tff(fact_3157_AND__upper1_H_H,axiom,
    ! [Ya: int,Z: int,Yaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ya),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Ya),Yaa)),Z) ) ) ).

% AND_upper1''
tff(fact_3158_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_3159_bit__imp__take__bit__positive,axiom,
    ! [Na: nat,M: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na)
       => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_3160_pi__less__4,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% pi_less_4
tff(fact_3161_pi__ge__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_3162_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat,A3: A] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
         => ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Na) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_3163_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M3)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),M3)
              <=> aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N2) ) )
         => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N2)
             => ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,N2),one_one(nat)))
              <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),N2) ) ) ) ).

% int_bit_bound
tff(fact_3164_pi__half__less__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_less_two
tff(fact_3165_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_le_two
tff(fact_3166_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = zero_zero(A) )
        <=> ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Na) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_3167_pi__half__gt__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_gt_zero
tff(fact_3168_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_ge_zero
tff(fact_3169_m2pi__less__pi,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))),pi) ).

% m2pi_less_pi
tff(fact_3170_arctan__ubound,axiom,
    ! [Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Ya)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arctan_ubound
tff(fact_3171_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),Na)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_3172_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Na)
          <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,minus_minus(A,A3),one_one(A))),Na)
              | ( Na = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_3173_OR__upper,axiom,
    ! [Xa: int,Na: nat,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Ya),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na))
         => aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xa),Ya)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),Na)) ) ) ) ).

% OR_upper
tff(fact_3174_minus__pi__half__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_3175_cot__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,cot(real),Xa)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_3176_cos__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( cos(real,Xa) = zero_zero(real) )
       => ? [N2: nat] :
            ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_3177_sin__zero__lemma,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( sin(real,Xa) = zero_zero(real) )
       => ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( Xa = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_3178_cot__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,cot(real),Xa)) ) ) ).

% cot_gt_zero
tff(fact_3179_arcsin__lbound,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Ya)) ) ) ).

% arcsin_lbound
tff(fact_3180_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_3181_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_3182_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_3183_sin__arcsin,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ( sin(real,aa(real,real,arcsin,Ya)) = Ya ) ) ) ).

% sin_arcsin
tff(fact_3184_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) = one_one(A) )
         => ( sin(A,Xa) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3185_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sin(A,Xa) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,Xa)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3186_sincos__principal__value,axiom,
    ! [Xa: real] :
    ? [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
      & ( sin(real,Y) = sin(real,Xa) )
      & ( cos(real,Y) = cos(real,Xa) ) ) ).

% sincos_principal_value
tff(fact_3187_sin__x__le__x,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),Xa) ) ).

% sin_x_le_x
tff(fact_3188_sin__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),one_one(real)) ).

% sin_le_one
tff(fact_3189_cos__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),one_one(real)) ).

% cos_le_one
tff(fact_3190_cos__arcsin__nonzero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xa)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3191_abs__sin__x__le__abs__x,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xa))),aa(real,real,abs_abs(real),Xa)) ).

% abs_sin_x_le_abs_x
tff(fact_3192_sin__cos__le1,axiom,
    ! [Xa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,Xa)),sin(real,Ya))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,Xa)),cos(real,Ya))))),one_one(real)) ).

% sin_cos_le1
tff(fact_3193_sin__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero
tff(fact_3194_sin__x__ge__neg__x,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xa)),sin(real,Xa)) ) ).

% sin_x_ge_neg_x
tff(fact_3195_sin__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_ge_zero
tff(fact_3196_sin__ge__minus__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),sin(real,Xa)) ).

% sin_ge_minus_one
tff(fact_3197_cos__inj__pi,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),pi)
           => ( ( cos(real,Xa) = cos(real,Ya) )
             => ( Xa = Ya ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_3198_cos__mono__le__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),cos(real,Ya))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),Xa) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_3199_cos__monotone__0__pi__le,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xa)),cos(real,Ya)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_3200_cos__ge__minus__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),cos(real,Xa)) ).

% cos_ge_minus_one
tff(fact_3201_abs__sin__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xa))),one_one(real)) ).

% abs_sin_le_one
tff(fact_3202_abs__cos__le__one,axiom,
    ! [Xa: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),cos(real,Xa))),one_one(real)) ).

% abs_cos_le_one
tff(fact_3203_arcsin__sin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arcsin,sin(real,Xa)) = Xa ) ) ) ).

% arcsin_sin
tff(fact_3204_arcsin__minus,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,Xa)) ) ) ) ).

% arcsin_minus
tff(fact_3205_arcsin__le__arcsin,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Ya)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3206_arcsin__eq__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),one_one(real))
       => ( ( aa(real,real,arcsin,Xa) = aa(real,real,arcsin,Ya) )
        <=> ( Xa = Ya ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3207_arcsin__le__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ) ) ).

% arcsin_le_mono
tff(fact_3208_cos__mono__less__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xa)),cos(real,Ya))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),Xa) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3209_cos__monotone__0__pi,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Xa)),cos(real,Ya)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3210_sin__eq__0__pi,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),pi)
       => ( ( sin(real,Xa) = zero_zero(real) )
         => ( Xa = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3211_sin__zero__pi__iff,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),pi)
     => ( ( sin(real,Xa) = zero_zero(real) )
      <=> ( Xa = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3212_cos__monotone__minus__pi__0_H,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Ya)),cos(real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_3213_arcsin,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Ya))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Ya)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( sin(real,aa(real,real,arcsin,Ya)) = Ya ) ) ) ) ).

% arcsin
tff(fact_3214_arcsin__pi,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Ya))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Ya)),pi)
          & ( sin(real,aa(real,real,arcsin,Ya)) = Ya ) ) ) ) ).

% arcsin_pi
tff(fact_3215_arcsin__le__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Xa)),Ya)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),sin(real,Ya)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3216_le__arcsin__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),aa(real,real,arcsin,Xa))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Ya)),Xa) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3217_sincos__total__pi,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),pi)
            & ( Xa = cos(real,T5) )
            & ( Ya = sin(real,T5) ) ) ) ) ).

% sincos_total_pi
tff(fact_3218_sin__cos__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xa))
     => ( sin(real,Xa) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,cos(real,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3219_cos__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xa)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3220_arcsin__less__arcsin,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Ya)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3221_sin__gt__zero__02,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero_02
tff(fact_3222_arcsin__less__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Xa)),aa(real,real,arcsin,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ) ) ).

% arcsin_less_mono
tff(fact_3223_cos__two__less__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_3224_cos__is__zero,axiom,
    ? [X3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X3) = zero_zero(real) )
      & ! [Y2: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( cos(real,Y2) = zero_zero(real) ) )
         => ( Y2 = X3 ) ) ) ).

% cos_is_zero
tff(fact_3225_cos__two__le__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_3226_cos__monotone__minus__pi__0,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,Ya)),cos(real,Xa)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3227_cos__total,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ? [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),pi)
            & ( cos(real,X3) = Ya )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),pi)
                  & ( cos(real,Y2) = Ya ) )
               => ( Y2 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_3228_sincos__total__pi__half,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
         => ? [T5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( Xa = cos(real,T5) )
              & ( Ya = sin(real,T5) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3229_sincos__total__2pi__le,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ? [T5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( Xa = cos(real,T5) )
          & ( Ya = sin(real,T5) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3230_sincos__total__2pi,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xa = cos(real,T5) )
               => ( Ya != sin(real,T5) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3231_sin__pi__divide__n__ge__0,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),Na)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3232_sin__gt__zero2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,Xa)) ) ) ).

% sin_gt_zero2
tff(fact_3233_sin__lt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),pi),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_3234_cos__double__less__one,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xa))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_3235_cos__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_gt_zero
tff(fact_3236_sin__monotone__2pi__le,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Ya)),sin(real,Xa)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3237_sin__mono__le__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),sin(real,Ya))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3238_sin__inj__pi,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( sin(real,Xa) = sin(real,Ya) )
             => ( Xa = Ya ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3239_sin__le__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),pi),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_3240_sin__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3241_sin__monotone__2pi,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Ya)),sin(real,Xa)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3242_sin__mono__less__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),sin(real,Xa)),sin(real,Ya))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3243_sin__total,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ? [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),X3)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X3) = Ya )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( sin(real,Y2) = Ya ) )
               => ( Y2 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_3244_cos__gt__zero__pi,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_gt_zero_pi
tff(fact_3245_cos__ge__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cos(real,Xa)) ) ) ).

% cos_ge_zero
tff(fact_3246_sin__pi__divide__n__gt__0,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),Na)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3247_arcsin__lt__bounded,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Ya))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arcsin,Ya)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3248_arcsin__bounded,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Ya))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Ya)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_3249_arcsin__ubound,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arcsin,Ya)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arcsin_ubound
tff(fact_3250_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xa)) != 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),bit0(one2))),Xa)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,tan(A),Xa)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_3251_sin__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( sin(real,Xa) = divide_divide(real,aa(real,real,tan(real),Xa),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_3252_cos__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( cos(real,Xa) = 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,power_power(real,aa(real,real,tan(real),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_3253_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(cos(real,T5),sin(real,T5)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3254_sin__arccos__abs,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),one_one(real))
     => ( sin(real,aa(real,real,arccos,Ya)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3255_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_3256_cos__arccos,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ( cos(real,aa(real,real,arccos,Ya)) = Ya ) ) ) ).

% cos_arccos
tff(fact_3257_arccos__le__arccos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Ya)),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_le_arccos
tff(fact_3258_arccos__le__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Xa)),aa(real,real,arccos,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),Xa) ) ) ) ).

% arccos_le_mono
tff(fact_3259_arccos__eq__iff,axiom,
    ! [Xa: real,Ya: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),one_one(real)) )
     => ( ( aa(real,real,arccos,Xa) = aa(real,real,arccos,Ya) )
      <=> ( Xa = Ya ) ) ) ).

% arccos_eq_iff
tff(fact_3260_arccos__lbound,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Ya)) ) ) ).

% arccos_lbound
tff(fact_3261_arccos__less__arccos,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Ya)),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_less_arccos
tff(fact_3262_arccos__less__mono,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Xa)),aa(real,real,arccos,Ya))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),Xa) ) ) ) ).

% arccos_less_mono
tff(fact_3263_arccos__ubound,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Ya)),pi) ) ) ).

% arccos_ubound
tff(fact_3264_arccos__cos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
       => ( aa(real,real,arccos,cos(real,Xa)) = Xa ) ) ) ).

% arccos_cos
tff(fact_3265_cos__arccos__abs,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Ya)),one_one(real))
     => ( cos(real,aa(real,real,arccos,Ya)) = Ya ) ) ).

% cos_arccos_abs
tff(fact_3266_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Theta)),pi)
     => ( aa(real,real,arccos,cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_3267_arccos__lt__bounded,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,arccos,Ya))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,arccos,Ya)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_3268_arccos__bounded,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Ya))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Ya)),pi) ) ) ) ).

% arccos_bounded
tff(fact_3269_sin__arccos__nonzero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xa)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3270_arccos__cos2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Xa)
       => ( aa(real,real,arccos,cos(real,Xa)) = aa(real,real,uminus_uminus(real),Xa) ) ) ) ).

% arccos_cos2
tff(fact_3271_arccos__minus,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,Xa)) ) ) ) ).

% arccos_minus
tff(fact_3272_tan__gt__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,tan(real),Xa)) ) ) ).

% tan_gt_zero
tff(fact_3273_lemma__tan__total,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Ya)
     => ? [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),X3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),aa(real,real,tan(real),X3)) ) ) ).

% lemma_tan_total
tff(fact_3274_lemma__tan__total1,axiom,
    ! [Ya: real] :
    ? [X3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),X3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X3) = Ya ) ) ).

% lemma_tan_total1
tff(fact_3275_tan__mono__lt__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Ya))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3276_tan__monotone_H,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),Xa)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Ya)),aa(real,real,tan(real),Xa)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3277_tan__monotone,axiom,
    ! [Ya: real,Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Ya)),aa(real,real,tan(real),Xa)) ) ) ) ).

% tan_monotone
tff(fact_3278_tan__total,axiom,
    ! [Ya: real] :
    ? [X3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),X3)
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X3) = Ya )
      & ! [Y2: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Y2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,tan(real),Y2) = Ya ) )
         => ( Y2 = X3 ) ) ) ).

% tan_total
tff(fact_3279_arccos,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Ya))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Ya)),pi)
          & ( cos(real,aa(real,real,arccos,Ya)) = Ya ) ) ) ) ).

% arccos
tff(fact_3280_arccos__minus__abs,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),Xa)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,Xa)) ) ) ).

% arccos_minus_abs
tff(fact_3281_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Ya: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Ya) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Ya)) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Ya))) ) ) ) ) ).

% add_tan_eq
tff(fact_3282_tan__total__pos,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
     => ? [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( aa(real,real,tan(real),X3) = Ya ) ) ) ).

% tan_total_pos
tff(fact_3283_tan__pos__pi2__le,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),Xa)) ) ) ).

% tan_pos_pi2_le
tff(fact_3284_tan__less__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,tan(real),Xa)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3285_tan__mono__le,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Ya)) ) ) ) ).

% tan_mono_le
tff(fact_3286_tan__mono__le__eq,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Ya),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,tan(real),Xa)),aa(real,real,tan(real),Ya))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),Ya) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3287_tan__bound__pi2,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),Xa))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3288_arctan,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Ya))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),arctan(Ya)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),arctan(Ya)) = Ya ) ) ).

% arctan
tff(fact_3289_arctan__tan,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( arctan(aa(real,real,tan(real),Xa)) = Xa ) ) ) ).

% arctan_tan
tff(fact_3290_arctan__unique,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),Xa) = Ya )
         => ( arctan(Ya) = Xa ) ) ) ) ).

% arctan_unique
tff(fact_3291_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Ya: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Ya) != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Ya))) = divide_divide(A,cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Ya))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3292_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Ya: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Ya) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,minus_minus(A,Xa),Ya)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,minus_minus(A,Xa),Ya)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Ya)),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),Xa)),aa(A,A,tan(A),Ya)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3293_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Ya: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( ( cos(A,Ya) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Ya)),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xa)),aa(A,A,tan(A),Ya)))) ) ) ) ) ) ).

% tan_add
tff(fact_3294_tan__total__pi4,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ? [Z2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z2) = Xa ) ) ) ).

% tan_total_pi4
tff(fact_3295_arccos__le__pi2,axiom,
    ! [Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Ya)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Ya),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,arccos,Ya)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arccos_le_pi2
tff(fact_3296_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),cos(A,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_3297_sin__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xa)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_3298_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3299_complete__linorder__sup__max,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ( sup_sup(A) = ord_max(A) ) ) ).

% complete_linorder_sup_max
tff(fact_3300_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(list($o),int,aa(int,fun(list($o),int),aa(fun($o,int),fun(int,fun(list($o),int)),groups4207007520872428315er_sum($o,int),zero_neq_one_of_bool(int)),aa(num,int,numeral_numeral(int),bit0(one2))),Bs)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(list($o),nat,size_size(list($o)),Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_3301_push__bit__of__Suc__0,axiom,
    ! [Na: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,Na),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na) ).

% push_bit_of_Suc_0
tff(fact_3302_set__removeAll,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(list(A),set(A),set2(A),removeAll(A,Xa,Xs)) = aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_3303_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,Na),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_3304_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A3: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,Na),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_3305_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_3306_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xa: real] :
          ( ( real_Vector_of_real(A,Xa) = zero_zero(A) )
        <=> ( Xa = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_3307_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_3308_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se4730199178511100633sh_bit(A,Na),A3))
        <=> ( ( Na != zero_zero(nat) )
            | aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% even_push_bit_iff
tff(fact_3309_length__removeAll__less__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,Xa,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_removeAll_less_eq
tff(fact_3310_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q5: nat,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),Q5)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(nat,Q5),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3311_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,Xa))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_3312_length__removeAll__less,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),removeAll(A,Xa,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_removeAll_less
tff(fact_3313_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Na)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_se4730199178511100633sh_bit(A,Na),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_3314_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B3: real,A3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,real_Vector_of_real(A,B3)),real_Vector_of_real(A,A3)))),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,B3),A3))) ) ).

% norm_of_real_diff
tff(fact_3315_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($o,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),Bs)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),Na) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3316_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Na: nat] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X3) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% inthall
tff(fact_3317_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat,Xa: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Deg,TreeList,Summary)),Xa) ) ) ) ).

% both_member_options_ding
tff(fact_3318_sub__num__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,bit1(L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(L))) ) ).

% sub_num_simps(3)
tff(fact_3319_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Na)
     => ( Deg = Na ) ) ).

% deg_deg_n
tff(fact_3320_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,Na)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Na)),TreeList2,S4) ) ).

% deg_SUcn_Node
tff(fact_3321_push__bit__nonnegative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4730199178511100633sh_bit(int,Na),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_3322_push__bit__negative__int__iff,axiom,
    ! [Na: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_se4730199178511100633sh_bit(int,Na),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_3323_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_3324_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Na)) = neg_numeral_sub(A,M,Na) ) ).

% diff_numeral_simps(1)
tff(fact_3325_sub__num__simps_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,bit0(K),bit0(L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(6)
tff(fact_3326_sub__num__simps_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,bit1(K),bit1(L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(9)
tff(fact_3327_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: 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),Na))) = neg_numeral_sub(A,M,Na) ) ).

% add_neg_numeral_simps(1)
tff(fact_3328_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: 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),Na)) = neg_numeral_sub(A,Na,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_3329_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Na: num] : aa(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),Na))) = neg_numeral_sub(A,Na,M) ) ).

% diff_numeral_simps(4)
tff(fact_3330_sub__num__simps_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,bit1(K),bit0(L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(8)
tff(fact_3331_sub__num__simps_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,bit0(K),bit1(L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(7)
tff(fact_3332_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(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_3333_diff__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,minus_minus(A,one_one(A)),aa(num,A,numeral_numeral(A),Na)) = neg_numeral_sub(A,one2,Na) ) ).

% diff_numeral_special(1)
tff(fact_3334_sub__num__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,bit1(K),one2) = aa(num,A,numeral_numeral(A),bit0(K)) ) ).

% sub_num_simps(5)
tff(fact_3335_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_3336_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_3337_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_3338_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: 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),Na)) = neg_numeral_sub(A,Na,one2) ) ).

% add_neg_numeral_special(4)
tff(fact_3339_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num] : aa(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_3340_diff__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(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),Na))) = neg_numeral_sub(A,Na,one2) ) ).

% diff_numeral_special(7)
tff(fact_3341_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(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_3342_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: $o,X22: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X22)) ).

% VEBT.distinct(1)
tff(fact_3343_VEBT_Oexhaust,axiom,
    ! [Ya: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Ya != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X222: $o] : Ya != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_3344_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs = Ys2 )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys2),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_3345_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
         => ? [X_13: A] : aa(A,$o,aa(nat,fun(A,$o),P,I4),X_13) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I4),aa(nat,A,nth(A,Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_3346_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys2) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys2),I2) ) )
       => ( Xs = Ys2 ) ) ) ).

% nth_equalityI
tff(fact_3347_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts,S3),Xa) = vEBT_Node(Info,zero_zero(nat),Ts,S3) ).

% vebt_insert.simps(2)
tff(fact_3348_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_3349_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,K,L) = aa(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_3350_nth__mem,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,aa(nat,A,nth(A,Xs),Na),aa(list(A),set(A),set2(A),Xs)) ) ).

% nth_mem
tff(fact_3351_list__ball__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X3: A] :
            ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X3) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% list_ball_nth
tff(fact_3352_in__set__conv__nth,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I4) = Xa ) ) ) ).

% in_set_conv_nth
tff(fact_3353_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xa: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P,Xa) ) ) ).

% all_nth_imp_all_set
tff(fact_3354_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_3355_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,M),K)),Na)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
        & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_3356_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),Xa) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) ).

% vebt_insert.simps(3)
tff(fact_3357_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,Na,M))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na) ) ) ).

% sub_non_negative
tff(fact_3358_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),neg_numeral_sub(A,Na,M)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Na),M) ) ) ).

% sub_non_positive
tff(fact_3359_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),neg_numeral_sub(A,Na,M))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na) ) ) ).

% sub_positive
tff(fact_3360_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: num,M: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),neg_numeral_sub(A,Na,M)),zero_zero(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less(num),Na),M) ) ) ).

% sub_negative
tff(fact_3361_nth__rotate1,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,rotate1(A,Xs)),Na) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,Na),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_3362_in__children__def,axiom,
    ! [Na: nat,TreeList: list(vEBT_VEBT),Xa: nat] :
      ( vEBT_V5917875025757280293ildren(Na,TreeList,Xa)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,Na))),vEBT_VEBT_low(Xa,Na)) ) ).

% in_children_def
tff(fact_3363_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X3,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Na) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,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_3364_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X3,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_1)
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,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_3365_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary)),Xa) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_3366_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary)),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
        & ( ( Xa = Mi )
          | ( Xa = Ma )
          | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ma)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_3367_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Deg)
     => ( ( Mi = Ma )
       => ( ! [X2: vEBT_VEBT] :
              ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
             => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) )
          & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_3368_insert__simp__mima,axiom,
    ! [Xa: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
      ( ( ( Xa = Mi )
        | ( Xa = Ma ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary) ) ) ) ).

% insert_simp_mima
tff(fact_3369_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_3370_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary)),Xa)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xa = Mi )
          | ( Xa = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_3371_VEBT__internal_OminNull_Ocases,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ( Xa != vEBT_Leaf($false,$false) )
     => ( ! [Uv2: $o] : Xa != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xa != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy)
           => ~ ! [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc) ) ) ) ) ).

% VEBT_internal.minNull.cases
tff(fact_3372_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Xa)),aa(nat,nat,suc,aa(nat,nat,suc,V2)),TreeList,Summary) ).

% vebt_insert.simps(4)
tff(fact_3373_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( ! [X3: A,Y: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y),R2)
         => member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y),S3) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S3) ) ).

% subrelI
tff(fact_3374_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R2: A,S3: B,R: set(product_prod(A,B)),S5: B] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,R2),S3),R)
     => ( ( S5 = S3 )
       => member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,R2),S5),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_3375_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),zero_zero(nat),Va3,Vb2),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_3376_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Ya: $o] :
      ( ( vEBT_VEBT_minNull(Xa)
      <=> (Ya) )
     => ( ( ( Xa = vEBT_Leaf($false,$false) )
         => ~ (Ya) )
       => ( ( ? [Uv2: $o] : Xa = vEBT_Leaf($true,(Uv2))
           => (Ya) )
         => ( ( ? [Uu2: $o] : Xa = vEBT_Leaf((Uu2),$true)
             => (Ya) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy)
               => ~ (Ya) )
             => ~ ( ? [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc)
                 => (Ya) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_3377_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : ~ vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_3378_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xa) ).

% vebt_member.simps(2)
tff(fact_3379_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy2: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy2)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_3380_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),zero_zero(nat),Uy2,Uz2)),Xa) ).

% vebt_member.simps(3)
tff(fact_3381_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xa)
     => ( ! [Uv2: $o] : Xa != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xa != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_3382_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz2: nat] : ~ vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy2),Uz2) ).

% VEBT_internal.membermima.simps(2)
tff(fact_3383_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xa)
     => ( ( Xa != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_3384_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,Xa: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V2),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xa) ).

% vebt_member.simps(4)
tff(fact_3385_invar__vebt_Osimps,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( vEBT_invar_vebt(A12,A23)
    <=> ( ( ? [A9: $o,B7: $o] : A12 = vEBT_Leaf((A9),(B7))
          & ( A23 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),A23,TreeList3,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X,N) )
            & vEBT_invar_vebt(Summary2,N)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_13) ) )
        | ? [TreeList3: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT] :
            ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),A23,TreeList3,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X,N) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) )
            & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13)
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_13) ) )
        | ? [TreeList3: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),A23,TreeList3,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X,N) )
            & vEBT_invar_vebt(Summary2,N)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_13) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A23))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
                 => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(Ma2,N)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(X,N)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma2) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi2),Ma2)),A23,TreeList3,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
               => vEBT_invar_vebt(X,N) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)) )
            & ( A23 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,suc,N)) )
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)))
               => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),X_13)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I4) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))
                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_13) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),A23))
            & ( ( Mi2 != Ma2 )
             => ! [I4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(Ma2,N)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(X,N)) )
                       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi2),X)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_3386_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( vEBT_invar_vebt(A12,A23)
     => ( ( ? [A5: $o,B5: $o] : A12 = vEBT_Leaf((A5),(B5))
         => ( A23 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat] :
              ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3) )
             => ( ( A23 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                     => vEBT_invar_vebt(X2,N2) )
                 => ( vEBT_invar_vebt(Summary3,M4)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                     => ( ( M4 = N2 )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M4) )
                         => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat] :
                ( ( A12 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList2,Summary3) )
               => ( ( A23 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                       => vEBT_invar_vebt(X2,N2) )
                   => ( vEBT_invar_vebt(Summary3,M4)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                       => ( ( M4 = aa(nat,nat,suc,N2) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M4) )
                           => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                  ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),Deg2,TreeList2,Summary3) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                         => vEBT_invar_vebt(X2,N2) )
                     => ( vEBT_invar_vebt(Summary3,M4)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                         => ( ( M4 = N2 )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M4) )
                             => ( ! [I3: nat] :
                                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_13)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I3) ) )
                               => ( ( ( Mi3 = Ma3 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) ) )
                                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                     => ~ ( ( Mi3 != Ma3 )
                                         => ! [I3: nat] :
                                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                             => ( ( ( vEBT_VEBT_high(Ma3,N2) = I3 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma3,N2)) )
                                                & ! [X2: nat] :
                                                    ( ( ( vEBT_VEBT_high(X2,N2) = I3 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X2,N2)) )
                                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X2)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Ma3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M4: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                    ( ( A12 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),Deg2,TreeList2,Summary3) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                           => vEBT_invar_vebt(X2,N2) )
                       => ( vEBT_invar_vebt(Summary3,M4)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4) )
                           => ( ( M4 = aa(nat,nat,suc,N2) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M4) )
                               => ( ! [I3: nat] :
                                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                     => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_13)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I3) ) )
                                 => ( ( ( Mi3 = Ma3 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                         => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_12) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi3),Ma3)
                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                       => ~ ( ( Mi3 != Ma3 )
                                           => ! [I3: nat] :
                                                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M4))
                                               => ( ( ( vEBT_VEBT_high(Ma3,N2) = I3 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma3,N2)) )
                                                  & ! [X2: nat] :
                                                      ( ( ( vEBT_VEBT_high(X2,N2) = I3 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X2,N2)) )
                                                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi3),X2)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Ma3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_3387_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X3,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Na )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Na) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,Na)) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,Na) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X3,Na)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X3)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_3388_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),Na: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
         => vEBT_invar_vebt(X3,Na) )
     => ( vEBT_invar_vebt(Summary,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,Na) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_13)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_1) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,Na) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,Na)) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,Na) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X3,Na)) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),X3)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_3389_insert__simp__norm,axiom,
    ! [Xa: nat,Deg: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Mi),Xa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Xa),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Xa,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_3390_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: list(vEBT_VEBT),Xa: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xa != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Xa) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Mi),Ma))),Deg,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary,vEBT_VEBT_high(Mi,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_3391_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q5: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q5),R2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),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),bit0(one2))),Q5)),one_one(A))),aa(A,A,minus_minus(A,R2),aa(num,A,numeral_numeral(A),L))),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),bit0(one2))),Q5)),R2)) ) ).

% divmod_step_eq
tff(fact_3392_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q5: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),product_Pair(A,A,Q5),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_3393_product__nth,axiom,
    ! [A: $tType,B: $tType,Na: nat,Xs: list(A),Ys2: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys2)),Na) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),divide_divide(nat,Na,aa(list(B),nat,size_size(list(B)),Ys2)))),aa(nat,B,nth(B,Ys2),modulo_modulo(nat,Na,aa(list(B),nat,size_size(list(B)),Ys2)))) ) ) ).

% product_nth
tff(fact_3394_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( list_update(A,Xs,I,Xa) = Xs ) ) ).

% list_update_beyond
tff(fact_3395_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xa)),I) = Xa ) ) ).

% nth_list_update_eq
tff(fact_3396_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_3397_less__by__empty,axiom,
    ! [A: $tType,A4: set(product_prod(A,A)),B2: set(product_prod(A,A))] :
      ( ( A4 = bot_bot(set(product_prod(A,A))) )
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),A4),B2) ) ).

% less_by_empty
tff(fact_3398_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A4: set(A),Xa: A,I: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A4)
     => ( member(A,Xa,A4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xa))),A4) ) ) ).

% set_update_subsetI
tff(fact_3399_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,D6: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),D6)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,Deg3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList2,Summary3)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_3400_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xa))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_3401_set__update__memI,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,Xa,aa(list(A),set(A),set2(A),list_update(A,Xs,Na,Xa))) ) ).

% set_update_memI
tff(fact_3402_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I,Xa) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = Xa ) ) ) ).

% list_update_same_conv
tff(fact_3403_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xa: A,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xa)),J) = $ite(I = J,Xa,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_3404_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B5: $o,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),X3)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Ux2)
       => ~ ! [Uy: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4)),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_3405_vebt__insert_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B5: $o,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),X3)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,zero_zero(nat),Ts2,S4)),X3)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4)),X3)
         => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3)),X3)
           => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)),X3) ) ) ) ) ).

% vebt_insert.cases
tff(fact_3406_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((Uu2),(Uv2))),Uw2)
     => ( ! [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT,Uz: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)),Uz)
       => ( ! [Mi3: nat,Ma3: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb)),X3)
         => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc)),X3)
           => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd)),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_3407_vebt__member_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: $o,B5: $o,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),X3)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),X3)
       => ( ! [V3: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy,Uz)),X3)
         => ( ! [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),X3)
           => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,X3: nat] : Xa != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)),X3) ) ) ) ) ).

% vebt_member.cases
tff(fact_3408_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit1(M),bit1(Na)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na),aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),bit1(M))),unique1321980374590559556d_step(A,bit1(Na),unique8689654367752047608divmod(A,bit1(M),bit0(bit1(Na))))) ) ).

% divmod_algorithm_code(8)
tff(fact_3409_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,bit0(M),bit1(Na)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Na),aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))),unique1321980374590559556d_step(A,bit1(Na),unique8689654367752047608divmod(A,bit0(M),bit0(bit1(Na))))) ) ).

% divmod_algorithm_code(7)
tff(fact_3410_neg__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q5: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B3,aa(int,product_prod(int,int),product_Pair(int,int,Q5),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),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3),aa(int,product_prod(int,int),product_Pair(int,int,Q5),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_3411_pos__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q5: int,R2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B3)
     => ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),product_Pair(int,int,Q5),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),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3),aa(int,product_prod(int,int),product_Pair(int,int,Q5),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_3412_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_3413_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_3414_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : unique8689654367752047608divmod(A,one2,bit0(Na)) = 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_3415_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Na: num] : unique8689654367752047608divmod(A,one2,bit1(Na)) = 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_3416_option_Osize__neq,axiom,
    ! [A: $tType,Xa: option(A)] : aa(option(A),nat,size_size(option(A)),Xa) != zero_zero(nat) ).

% option.size_neq
tff(fact_3417_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q5: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q5),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),Q5)),R2) )
        & $ite(
            aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R2)
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),R2),L) ),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)),
              ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),R2)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R2),zero_zero(int)) ),
              Q5 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_3418_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q5: int] :
      ( ( sgn_sgn(int,R2) = sgn_sgn(int,L) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R2)),aa(int,int,abs_abs(int),L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q5),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_3419_eucl__rel__int_Osimps,axiom,
    ! [A12: int,A23: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A12,A23,A32)
    <=> ( ? [K3: int] :
            ( ( A12 = K3 )
            & ( A23 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),K3) ) )
        | ? [L3: int,K3: int,Q6: int] :
            ( ( A12 = K3 )
            & ( A23 = L3 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q6),zero_zero(int)) )
            & ( L3 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q6),L3) ) )
        | ? [R5: int,L3: int,K3: int,Q6: int] :
            ( ( A12 = K3 )
            & ( A23 = L3 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q6),R5) )
            & ( sgn_sgn(int,R5) = sgn_sgn(int,L3) )
            & aa(int,$o,aa(int,fun(int,$o),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),Q6),L3)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_3420_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A12,A23,A32)
     => ( ( ( A23 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),A12) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q3),zero_zero(int)) )
             => ( ( A23 != zero_zero(int) )
               => ( A12 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A23) ) ) )
         => ~ ! [R3: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3) )
               => ( ( sgn_sgn(int,R3) = sgn_sgn(int,A23) )
                 => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),A23))
                   => ( A12 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A23)),R3) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_3421_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Na: num] :
          unique8689654367752047608divmod(A,M,Na) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less(num),M),Na),aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),M)),unique1321980374590559556d_step(A,Na,unique8689654367752047608divmod(A,M,bit0(Na)))) ) ).

% divmod_divmod_step
tff(fact_3422_option_Osize_I4_J,axiom,
    ! [A: $tType,X23: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X23)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_3423_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,nat),X23: A] : size_option(A,Xa,aa(A,option(A),some(A),X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_3424_vebt__insert_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Ya )
     => ( ! [A5: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B5)) )
           => ( Ya != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B5)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A5),$true),vEBT_Leaf((A5),(B5))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts2,S4) )
             => ( Ya != vEBT_Node(Info2,zero_zero(nat),Ts2,S4) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4) )
               => ( Ya != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4) ) )
           => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3) )
                 => ( Ya != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3) ) )
             => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3) )
                   => ( Ya != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),Mi3,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi3 )
                                  | ( Xaa = Ma3 ) ) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),
                                    product_Pair(nat,nat,
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),Xaa,Mi3)),
                                    aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary3,h),Summary3)),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_3425_vebt__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
      <=> (Ya) )
     => ( ! [A5: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B5)) )
           => ( (Ya)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B5),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => (Ya) )
         => ( ( ? [V3: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy,Uz)
             => (Ya) )
           => ( ( ? [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)
               => (Ya) )
             => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary3: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)
                   => ( (Ya)
                    <=> ~ $ite(
                            Xaa = Mi3,
                            $true,
                            $ite(
                              Xaa = Ma3,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_3426_vebt__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( ! [A5: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B5)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A5),
                $ite(Xaa = one_one(nat),(B5),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [V3: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy,Uz)
           => ( ! [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xa != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)
             => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary3: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)
                   => $ite(
                        Xaa = Mi3,
                        $true,
                        $ite(
                          Xaa = Ma3,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_3427_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary),Xa) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Mi,Xa),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xa = Mi )
                | ( Xa = Ma ) ) ),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                aa(nat,product_prod(nat,nat),
                  product_Pair(nat,nat,
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),Xa,Mi)),
                  aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),Summary)),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_3428_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_3429_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
        & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3430_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
        | aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),Q)) )
     => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_3431_finite__Collect__subsets,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(set(A)),$o,finite_finite2(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ac(set(A),fun(set(A),$o),A4))) ) ).

% finite_Collect_subsets
tff(fact_3432_finite__nth__roots,axiom,
    ! [Na: nat,C3: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(set(complex),$o,finite_finite2(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_ad(nat,fun(complex,fun(complex,$o)),Na),C3))) ) ).

% finite_nth_roots
tff(fact_3433_singleton__conv,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ae(A,fun(A,$o),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ).

% singleton_conv
tff(fact_3434_singleton__conv2,axiom,
    ! [A: $tType,A3: A] : aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),fequal(A),A3)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_3435_finite__Collect__less__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),K))) ).

% finite_Collect_less_nat
tff(fact_3436_finite__Collect__le__nat,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ag(nat,fun(nat,$o)),K))) ).

% finite_Collect_le_nat
tff(fact_3437_card__Collect__less__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Na))) = Na ).

% card_Collect_less_nat
tff(fact_3438_finite__interval__int1,axiom,
    ! [A3: int,B3: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ah(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int1
tff(fact_3439_finite__interval__int4,axiom,
    ! [A3: int,B3: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ai(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int4
tff(fact_3440_card__Collect__le__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ag(nat,fun(nat,$o)),Na))) = aa(nat,nat,suc,Na) ).

% card_Collect_le_nat
tff(fact_3441_finite__interval__int3,axiom,
    ! [A3: int,B3: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_aj(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int3
tff(fact_3442_finite__interval__int2,axiom,
    ! [A3: int,B3: int] : aa(set(int),$o,finite_finite2(int),aa(fun(int,$o),set(int),collect(int),aa(int,fun(int,$o),aTP_Lamp_ak(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int2
tff(fact_3443_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),S))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S) ) ).

% pred_subset_eq2
tff(fact_3444_inf__Int__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),R),S)) ) ).

% inf_Int_eq2
tff(fact_3445_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X: A,Xa3: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X),Xa3),R)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X),Xa3),S) )
    <=> ( R = S ) ) ).

% pred_equals_eq2
tff(fact_3446_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_3447_sup__Un__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(A,B)),X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),sup_sup(fun(A,fun(B,$o))),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),R)),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),S)),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),R),S)) ) ).

% sup_Un_eq2
tff(fact_3448_Collect__conv__if,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_3449_Collect__conv__if2,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_3450_empty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ao(A,$o)) ).

% empty_def
tff(fact_3451_Collect__imp__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_imp_eq
tff(fact_3452_Collect__disj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_disj_eq
tff(fact_3453_sup__set__def,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2))) ).

% sup_set_def
tff(fact_3454_Un__def,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aq(set(A),fun(set(A),fun(A,$o)),A4),B2)) ).

% Un_def
tff(fact_3455_sup__Un__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A),X2: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),sup_sup(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X2)
    <=> member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),R),S)) ) ).

% sup_Un_eq
tff(fact_3456_insert__def,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ae(A,fun(A,$o),A3))),B2) ).

% insert_def
tff(fact_3457_insert__Collect,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(fun(A,$o),set(A),collect(A),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ar(A,fun(fun(A,$o),fun(A,$o)),A3),P)) ).

% insert_Collect
tff(fact_3458_insert__compr,axiom,
    ! [A: $tType,A3: A,B2: set(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),B2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_as(A,fun(set(A),fun(A,$o)),A3),B2)) ).

% insert_compr
tff(fact_3459_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_at(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_3460_card__roots__unity__eq,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_au(nat,fun(complex,$o),Na))) = Na ) ) ).

% card_roots_unity_eq
tff(fact_3461_card__nth__roots,axiom,
    ! [C3: complex,Na: nat] :
      ( ( C3 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( aa(set(complex),nat,finite_card(complex),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_av(complex,fun(nat,fun(complex,$o)),C3),Na))) = Na ) ) ) ).

% card_nth_roots
tff(fact_3462_set__diff__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),B2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aw(set(A),fun(set(A),fun(A,$o)),A4),B2)) ).

% set_diff_eq
tff(fact_3463_minus__set__def,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),B2) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),minus_minus(fun(A,$o),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2))) ).

% minus_set_def
tff(fact_3464_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B),R: fun(A,fun(B,$o))] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( ! [X3: A] :
              ( member(A,X3,A4)
             => ? [Xa2: B] :
                  ( member(B,Xa2,B2)
                  & aa(B,$o,aa(A,fun(B,$o),R,X3),Xa2) ) )
         => ? [X3: B] :
              ( member(B,X3,B2)
              & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ax(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A4),R),X3))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3465_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% not_finite_existsD
tff(fact_3466_inf__Int__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A),X2: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X2)
    <=> member(A,X2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),R),S)) ) ).

% inf_Int_eq
tff(fact_3467_Int__def,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ay(set(A),fun(set(A),fun(A,$o)),A4),B2)) ).

% Int_def
tff(fact_3468_Int__Collect,axiom,
    ! [A: $tType,Xa: A,A4: set(A),P: fun(A,$o)] :
      ( member(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))
    <=> ( member(A,Xa,A4)
        & aa(A,$o,P,Xa) ) ) ).

% Int_Collect
tff(fact_3469_inf__set__def,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2))) ).

% inf_set_def
tff(fact_3470_Collect__conj__eq,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(fun(A,$o),set(A),collect(A),P)),aa(fun(A,$o),set(A),collect(A),Q)) ).

% Collect_conj_eq
tff(fact_3471_uminus__set__def,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4))) ).

% uminus_set_def
tff(fact_3472_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P)) ).

% Collect_neg_eq
tff(fact_3473_Compl__eq,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ba(set(A),fun(A,$o),A4)) ).

% Compl_eq
tff(fact_3474_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bb(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_3475_less__set__def,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2)) ) ).

% less_set_def
tff(fact_3476_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bc(A,fun(A,$o),A3))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bc(A,fun(A,$o),B3)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3)
            & ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),A3) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3477_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bc(A,fun(A,$o),A3))),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bc(A,fun(A,$o),B3)))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3) ) ) ).

% subset_divisors_dvd
tff(fact_3478_less__eq__set__def,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),B2)) ) ).

% less_eq_set_def
tff(fact_3479_Collect__restrict,axiom,
    ! [A: $tType,X4: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),X4),P))),X4) ).

% Collect_restrict
tff(fact_3480_prop__restrict,axiom,
    ! [A: $tType,Xa: A,Z6: set(A),X4: set(A),P: fun(A,$o)] :
      ( member(A,Xa,Z6)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z6),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),X4),P)))
       => aa(A,$o,P,Xa) ) ) ).

% prop_restrict
tff(fact_3481_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S: set(A)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),R)),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R),S) ) ).

% pred_subset_eq
tff(fact_3482_Collect__subset,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))),A4) ).

% Collect_subset
tff(fact_3483_finite__less__ub,axiom,
    ! [F2: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),aa(nat,nat,F2,N2))
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_be(fun(nat,nat),fun(nat,fun(nat,$o)),F2),U))) ) ).

% finite_less_ub
tff(fact_3484_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa2: A] :
          aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Xa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa2),Xa2,X2) ) ).

% max_def_raw
tff(fact_3485_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] :
          aa(num,A,numeral_numeral(A),bit0(Na)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),Na),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m) ) ) ).

% numeral_code(2)
tff(fact_3486_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = aa(fun(nat,$o),set(nat),collect(nat),vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_3487_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_bf(A,fun(A,fun(A,$o)),A3),B3))) ) ).

% finite_int_segment
tff(fact_3488_nat__less__as__int,axiom,
    ! [X2: nat,Xa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X2),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa2)) ) ).

% nat_less_as_int
tff(fact_3489_nat__leq__as__int,axiom,
    ! [X2: nat,Xa2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Xa2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa2)) ) ).

% nat_leq_as_int
tff(fact_3490_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Na: num] :
          aa(num,A,numeral_numeral(A),bit1(Na)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),Na),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m)),one_one(A)) ) ) ).

% numeral_code(3)
tff(fact_3491_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bg(A,fun(A,$o),A3))) ) ).

% finite_abs_int_segment
tff(fact_3492_n__subsets,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(set(A)),nat,finite_card(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(nat,fun(set(A),$o),aTP_Lamp_bh(set(A),fun(nat,fun(set(A),$o)),A4),K))) = binomial(aa(set(A),nat,finite_card(A),A4),K) ) ) ).

% n_subsets
tff(fact_3493_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bi(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_3494_card__less,axiom,
    ! [M5: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M5)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bj(set(nat),fun(nat,fun(nat,$o)),M5),I))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_3495_card__less__Suc,axiom,
    ! [M5: set(nat),I: nat] :
      ( member(nat,zero_zero(nat),M5)
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(set(nat),fun(nat,fun(nat,$o)),M5),I)))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bj(set(nat),fun(nat,fun(nat,$o)),M5),I))) ) ) ).

% card_less_Suc
tff(fact_3496_card__less__Suc2,axiom,
    ! [M5: set(nat),I: nat] :
      ( ~ member(nat,zero_zero(nat),M5)
     => ( aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bk(set(nat),fun(nat,fun(nat,$o)),M5),I))) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_bj(set(nat),fun(nat,fun(nat,$o)),M5),I))) ) ) ).

% card_less_Suc2
tff(fact_3497_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bl(nat,fun(A,$o),Na))) ) ) ).

% finite_roots_unity
tff(fact_3498_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_bl(nat,fun(A,$o),Na)))),Na) ) ) ).

% card_roots_unity
tff(fact_3499_finite__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bm(set(A),fun(nat,fun(list(A),$o)),A4),Na))) ) ).

% finite_lists_length_eq
tff(fact_3500_card__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bm(set(A),fun(nat,fun(list(A),$o)),A4),Na))) = aa(nat,nat,power_power(nat,aa(set(A),nat,finite_card(A),A4)),Na) ) ) ).

% card_lists_length_eq
tff(fact_3501_diff__nat__eq__if,axiom,
    ! [Z: int,Z4: int] :
      aa(nat,nat,minus_minus(nat,aa(int,nat,nat2,Z)),aa(int,nat,nat2,Z4)) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),Z4),zero_zero(int)),
        aa(int,nat,nat2,Z),
        $let(
          d: int,
          d:= aa(int,int,minus_minus(int,Z),Z4),
          $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),d),zero_zero(int)),zero_zero(nat),aa(int,nat,nat2,d)) ) ) ).

% diff_nat_eq_if
tff(fact_3502_finite__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bn(set(A),fun(nat,fun(list(A),$o)),A4),Na))) ) ).

% finite_lists_length_le
tff(fact_3503_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] :
          gbinomial(A,A3,K) = $ite(K = zero_zero(nat),one_one(A),divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_bo(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,minus_minus(nat,K),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_3504_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option(product_prod(nat,nat)),V2: nat,TreeList: list(vEBT_VEBT),S3: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy2,aa(nat,nat,suc,V2),TreeList,S3),Xa)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3505_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V2),TreeList,Vd2),Xa)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3506_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Xa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),TreeList,Summary)),Xa)
    <=> $ite(
          Xa = Mi,
          $true,
          $ite(
            Xa = Ma,
            $true,
            $ite(
              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Mi),
              $false,
              $ite(
                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma),Xa),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_3507_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V2: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),aa(nat,nat,suc,V2),TreeList,Vc2),Xa)
    <=> ( ( Xa = Mi )
        | ( Xa = Ma )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3508_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( vEBT_V5719532721284313246member(Xa,Xaa)
      <=> (Ya) )
     => ( ! [A5: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B5)) )
           => ( (Ya)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B5),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => (Ya) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S4: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4)
               => ( (Ya)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3509_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xa,Xaa)
     => ( ! [A5: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B5)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A5),
                  $ite(Xaa = one_one(nat),(B5),$false) ) )
       => ~ ! [Uy: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S4: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3510_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xa,Xaa)
     => ( ! [A5: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B5)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A5),
                $ite(Xaa = one_one(nat),(B5),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xa != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S4: vEBT_VEBT] : Xa = vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3511_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xa,Xaa)
     => ( ! [Mi3: nat,Ma3: nat] :
            ( ? [Va2: list(vEBT_VEBT),Vb: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb)
           => ~ ( ( Xaa = Mi3 )
                | ( Xaa = Ma3 ) ) )
       => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc)
             => ~ ( ( Xaa = Mi3 )
                  | ( Xaa = Ma3 )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3512_vebt__member_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( ! [A5: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B5)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A5),
                  $ite(Xaa = one_one(nat),(B5),$false) ) )
       => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary3: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)
             => ~ $ite(
                    Xaa = Mi3,
                    $true,
                    $ite(
                      Xaa = Ma3,
                      $true,
                      $ite(
                        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),
                        $false,
                        $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_3513_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,nat)] : size_option(A,Xa,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_3514_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xa,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xa != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va2: list(vEBT_VEBT),Vb: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb)
               => ( ( Xaa = Mi3 )
                  | ( Xaa = Ma3 ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc)
                 => ( ( Xaa = Mi3 )
                    | ( Xaa = Ma3 )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3515_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( vEBT_VEBT_membermima(Xa,Xaa)
      <=> (Ya) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => (Ya) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)
           => (Ya) )
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va2: list(vEBT_VEBT),Vb: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb)
               => ( (Ya)
                <=> ~ ( ( Xaa = Mi3 )
                      | ( Xaa = Ma3 ) ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc)
                 => ( (Ya)
                  <=> ~ ( ( Xaa = Mi3 )
                        | ( Xaa = Ma3 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd)
                   => ( (Ya)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3516_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(
            K = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
              aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
                $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_3517_monoseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_bp(real,fun(nat,real),Xa)) ) ).

% monoseq_arctan_series
tff(fact_3518_vebt__insert_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xa,Xaa) = Ya )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [A5: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B5)) )
             => ( ( Ya = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B5)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A5),$true),vEBT_Leaf((A5),(B5))) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,zero_zero(nat),Ts2,S4) )
               => ( ( Ya = vEBT_Node(Info2,zero_zero(nat),Ts2,S4) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,zero_zero(nat),Ts2,S4)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4) )
                 => ( ( Ya = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S4)),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3) )
                   => ( ( Ya = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary3)),Xaa)) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3) )
                     => ( ( Ya = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),Mi3,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi3 )
                                      | ( Xaa = Ma3 ) ) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                      aa(nat,product_prod(nat,nat),
                                        product_Pair(nat,nat,
                                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),Xaa,Mi3)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),xn),Ma3))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary3,h),Summary3)),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_3519_ln__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),Xa) = suminf(real,aTP_Lamp_bq(real,fun(nat,real),Xa)) ) ) ) ).

% ln_series
tff(fact_3520_arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => ( arctan(Xa) = suminf(real,aTP_Lamp_br(real,fun(nat,real),Xa)) ) ) ).

% arctan_series
tff(fact_3521_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ! [X3: A,Y: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P,X3),Y)
         => aa(B,$o,aa(A,fun(B,$o),Q,X3),Y) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).

% predicate2I
tff(fact_3522_predicate1I,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
         => aa(A,$o,Q,X3) )
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q) ) ).

% predicate1I
tff(fact_3523_inf1I,axiom,
    ! [A: $tType,A4: fun(A,$o),Xa: A,B2: fun(A,$o)] :
      ( aa(A,$o,A4,Xa)
     => ( aa(A,$o,B2,Xa)
       => aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B2),Xa) ) ) ).

% inf1I
tff(fact_3524_inf2I,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),Xa: A,Ya: B,B2: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),A4,Xa),Ya)
     => ( aa(B,$o,aa(A,fun(B,$o),B2,Xa),Ya)
       => aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B2),Xa),Ya) ) ) ).

% inf2I
tff(fact_3525_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F2: fun(nat,A)] : suminf(A,aTP_Lamp_bs(fun(nat,A),fun(nat,A),F2)) = aa(nat,A,F2,zero_zero(nat)) ) ).

% powser_zero
tff(fact_3526_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xa: A,Ya: B,Q: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Xa),Ya)
     => ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
       => aa(B,$o,aa(A,fun(B,$o),Q,Xa),Ya) ) ) ).

% rev_predicate2D
tff(fact_3527_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Xa: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,Xa)
     => ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
       => aa(A,$o,Q,Xa) ) ) ).

% rev_predicate1D
tff(fact_3528_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),Xa: A,Ya: B] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
     => ( aa(B,$o,aa(A,fun(B,$o),P,Xa),Ya)
       => aa(B,$o,aa(A,fun(B,$o),Q,Xa),Ya) ) ) ).

% predicate2D
tff(fact_3529_predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xa: A] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
     => ( aa(A,$o,P,Xa)
       => aa(A,$o,Q,Xa) ) ) ).

% predicate1D
tff(fact_3530_inf2D2,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),B2: fun(A,fun(B,$o)),Xa: A,Ya: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B2),Xa),Ya)
     => aa(B,$o,aa(A,fun(B,$o),B2,Xa),Ya) ) ).

% inf2D2
tff(fact_3531_inf2D1,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),B2: fun(A,fun(B,$o)),Xa: A,Ya: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B2),Xa),Ya)
     => aa(B,$o,aa(A,fun(B,$o),A4,Xa),Ya) ) ).

% inf2D1
tff(fact_3532_inf1D2,axiom,
    ! [A: $tType,A4: fun(A,$o),B2: fun(A,$o),Xa: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B2),Xa)
     => aa(A,$o,B2,Xa) ) ).

% inf1D2
tff(fact_3533_inf1D1,axiom,
    ! [A: $tType,A4: fun(A,$o),B2: fun(A,$o),Xa: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B2),Xa)
     => aa(A,$o,A4,Xa) ) ).

% inf1D1
tff(fact_3534_inf2E,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o)),B2: fun(A,fun(B,$o)),Xa: A,Ya: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),inf_inf(fun(A,fun(B,$o))),A4),B2),Xa),Ya)
     => ~ ( aa(B,$o,aa(A,fun(B,$o),A4,Xa),Ya)
         => ~ aa(B,$o,aa(A,fun(B,$o),B2,Xa),Ya) ) ) ).

% inf2E
tff(fact_3535_inf1E,axiom,
    ! [A: $tType,A4: fun(A,$o),B2: fun(A,$o),Xa: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),inf_inf(fun(A,$o)),A4),B2),Xa)
     => ~ ( aa(A,$o,A4,Xa)
         => ~ aa(A,$o,B2,Xa) ) ) ).

% inf1E
tff(fact_3536_monoseq__realpow,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => topological_monoseq(real,power_power(real,Xa)) ) ) ).

% monoseq_realpow
tff(fact_3537_vebt__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
      <=> (Ya) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [A5: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B5)) )
             => ( ( (Ya)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B5),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ (Ya)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy,Uz) )
                 => ( ~ (Ya)
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy,Uz)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc) )
                   => ( ~ (Ya)
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),Xaa)) ) )
               => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3) )
                     => ( ( (Ya)
                        <=> $ite(
                              Xaa = Mi3,
                              $true,
                              $ite(
                                Xaa = Ma3,
                                $true,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),
                                  $false,
                                  $ite(
                                    aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_3538_vebt__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [A5: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B5)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B5),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) )
           => ( ! [V3: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy,Uz) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy,Uz)),Xaa)) )
             => ( ! [V3: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),Xaa)) )
               => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)),Xaa))
                       => $ite(
                            Xaa = Mi3,
                            $true,
                            $ite(
                              Xaa = Ma3,
                              $true,
                              $ite(
                                aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),
                                $false,
                                $ite(
                                  aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_3539_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( vEBT_V5719532721284313246member(Xa,Xaa)
      <=> (Ya) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [A5: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B5)) )
             => ( ( (Ya)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B5),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ (Ya)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)),Xaa)) ) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4) )
                 => ( ( (Ya)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3540_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [A5: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B5)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B5),$false) ) ) )
         => ~ ! [Uy: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3541_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [A5: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B5)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A5),
                    $ite(Xaa = one_one(nat),(B5),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) )
           => ~ ! [Uy: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S4: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy,aa(nat,nat,suc,V3),TreeList2,S4)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3542_vebt__member_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xa),Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [A5: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B5)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf((A5),(B5))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A5),
                      $ite(Xaa = one_one(nat),(B5),$false) ) ) )
         => ~ ! [Mi3: nat,Ma3: nat,Va: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList2,Summary3)),Xaa))
                 => ~ $ite(
                        Xaa = Mi3,
                        $true,
                        $ite(
                          Xaa = Ma3,
                          $true,
                          $ite(
                            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xaa),Mi3),
                            $false,
                            $ite(
                              aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ma3),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_3543_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ aa(product_prod(vEBT_VEBT,nat),$o,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))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)),Xaa)) )
           => ( ! [Mi3: nat,Ma3: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb)),Xaa))
                   => ( ( Xaa = Mi3 )
                      | ( Xaa = Ma3 ) ) ) )
             => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc) )
                   => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc)),Xaa))
                     => ( ( Xaa = Mi3 )
                        | ( Xaa = Ma3 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,V3),TreeList2,Vd)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3544_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( vEBT_VEBT_membermima(Xa,Xaa)
      <=> (Ya) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Ya)
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,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))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy) )
               => ( ~ (Ya)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy)),Xaa)) ) )
           => ( ! [Mi3: nat,Ma3: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb) )
                 => ( ( (Ya)
                    <=> ( ( Xaa = Mi3 )
                        | ( Xaa = Ma3 ) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb)),Xaa)) ) )
             => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc) )
                   => ( ( (Ya)
                      <=> ( ( Xaa = Mi3 )
                          | ( Xaa = Ma3 )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc)),Xaa)) ) )
               => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd) )
                     => ( ( (Ya)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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,V3),TreeList2,Vd)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3545_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => ( suminf(A,power_power(A,C3)) = divide_divide(A,one_one(A),aa(A,A,minus_minus(A,one_one(A)),C3)) ) ) ) ).

% suminf_geometric
tff(fact_3546_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [Mi3: nat,Ma3: nat,Va2: list(vEBT_VEBT),Vb: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),zero_zero(nat),Va2,Vb)),Xaa))
               => ~ ( ( Xaa = Mi3 )
                    | ( Xaa = Ma3 ) ) ) )
         => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList2,Vc)),Xaa))
                 => ~ ( ( Xaa = Mi3 )
                      | ( Xaa = Ma3 )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,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,V3),TreeList2,Vd)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3547_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_bt(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_3548_summable__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => summable(real,aTP_Lamp_br(real,fun(nat,real),Xa)) ) ).

% summable_arctan_series
tff(fact_3549_vebt__buildup_Oelims,axiom,
    ! [Xa: nat,Ya: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Ya )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Ya != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Ya != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va: nat] :
                ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
               => ( Ya != $ite(
                      aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                      $let(
                        half: nat,
                        half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3550_accp__subset,axiom,
    ! [A: $tType,R1: fun(A,fun(A,$o)),R22: fun(A,fun(A,$o))] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R1),R22)
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),accp(A,R22)),accp(A,R1)) ) ).

% accp_subset
tff(fact_3551_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xa: A,M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xa)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),
            zero_zero(A),
            $ite(Xa = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat))),M)),divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,Xa),M)),aa(nat,A,power_power(A,Xa),aa(nat,nat,suc,Na))),aa(A,A,minus_minus(A,one_one(A)),Xa))) ) ) ).

% sum_gp
tff(fact_3552_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xa: fun(A,B),Ya: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bu(set(A),fun(fun(A,B),fun(A,$o)),I5),Xa)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bu(set(A),fun(fun(A,B),fun(A,$o)),I5),Ya)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bv(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xa),Ya))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_3553_intind,axiom,
    ! [A: $tType,I: nat,Na: nat,P: fun(A,$o),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(A,$o,P,Xa)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Na,Xa)),I)) ) ) ).

% intind
tff(fact_3554_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,Xa: A,Na: nat,Ya: A] :
      ( ( replicate(A,M,Xa) = replicate(A,Na,Ya) )
    <=> ( ( M = Na )
        & ( ( M != zero_zero(nat) )
         => ( Xa = Ya ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3555_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_bw(B,A)),A4) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_3556_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_bx(nat,A)) ) ).

% summable_zero
tff(fact_3557_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_by(nat,fun(fun(nat,A),fun(nat,A)),I),F2)) ) ).

% summable_single
tff(fact_3558_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_3559_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F3: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),F3)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),F3) = zero_zero(B) )
          <=> ! [X: A] :
                ( member(A,X,F3)
               => ( aa(A,B,F2,X) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_3560_sum_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_3561_in__set__replicate,axiom,
    ! [A: $tType,Xa: A,Na: nat,Ya: A] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),replicate(A,Na,Ya)))
    <=> ( ( Xa = Ya )
        & ( Na != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_3562_Bex__set__replicate,axiom,
    ! [A: $tType,Na: nat,A3: A,P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),replicate(A,Na,A3)))
          & aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,A3)
        & ( Na != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_3563_Ball__set__replicate,axiom,
    ! [A: $tType,Na: nat,A3: A,P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),replicate(A,Na,A3)))
         => aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,A3)
        | ( Na = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_3564_nth__replicate,axiom,
    ! [A: $tType,I: nat,Na: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(nat,A,nth(A,replicate(A,Na,Xa)),I) = Xa ) ) ).

% nth_replicate
tff(fact_3565_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( 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_bz(A,fun(fun(A,B),fun(A,B)),A3),B3)),S) = $ite(member(A,A3,S),aa(A,B,B3,A3),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_3566_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( 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_ca(A,fun(fun(A,B),fun(A,B)),A3),B3)),S) = $ite(member(A,A3,S),aa(A,B,B3,A3),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_3567_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A4: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cb(fun(B,A),fun(B,A),F2)),A4)) ) ).

% sum_abs
tff(fact_3568_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(A,fun(fun(nat,A),fun(nat,A)),C3),F2))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_cmult_iff
tff(fact_3569_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(nat,A),C3: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cd(fun(nat,A),fun(A,fun(nat,A)),F2),C3))
        <=> ( ( C3 = zero_zero(A) )
            | summable(A,F2) ) ) ) ).

% summable_divide_iff
tff(fact_3570_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ce(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2)) ) ) ).

% summable_If_finite
tff(fact_3571_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A4: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A4)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cf(set(nat),fun(fun(nat,A),fun(nat,A)),A4),F2)) ) ) ).

% summable_If_finite_set
tff(fact_3572_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4)) ) ) ) ) ).

% sum.insert
tff(fact_3573_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F2: fun(B,A),A4: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aTP_Lamp_cb(fun(B,A),fun(B,A),F2)),A4)) ) ).

% sum_abs_ge_zero
tff(fact_3574_set__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( Na != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Na,Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_3575_sum__mult__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A4: set(A),F2: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_cg(fun(A,B),fun(fun(A,$o),fun(A,B)),F2),P)),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_mult_of_bool_eq
tff(fact_3576_sum__of__bool__mult__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A4: set(A),P: fun(A,$o),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_ch(fun(A,$o),fun(fun(A,B),fun(A,B)),P),F2)),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))) ) ) ) ).

% sum_of_bool_mult_eq
tff(fact_3577_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( summable(A,power_power(A,C3))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_3578_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: 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,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M),zero_zero(A),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,Na))),aa(nat,A,G,aa(nat,nat,suc,Na)))) ) ).

% sum.cl_ivl_Suc
tff(fact_3579_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C3: fun(nat,A),A4: set(nat)] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ci(fun(nat,A),fun(nat,A),C3)),A4) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A4)
            & member(nat,zero_zero(nat),A4) ),
            aa(nat,A,C3,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_3580_sum__of__bool__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [A4: set(A),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aTP_Lamp_cj(fun(A,$o),fun(A,B),P)),A4) = aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))) ) ) ) ) ).

% sum_of_bool_eq
tff(fact_3581_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C3: fun(nat,A),D2: fun(nat,A),A4: set(nat)] :
          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_ck(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C3),D2)),A4) = $ite(
            ( aa(set(nat),$o,finite_finite2(nat),A4)
            & member(nat,zero_zero(nat),A4) ),
            divide_divide(A,aa(nat,A,C3,zero_zero(nat)),aa(nat,A,D2,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_3582_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C3: A] :
          ( summable(A,aTP_Lamp_cl(A,fun(nat,A),C3))
        <=> ( C3 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_3583_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(A),B2: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),$o,finite_finite2(B),B2)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_cn(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B2),G),R)),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_cp(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A4),G),R)),B2) ) ) ) ) ).

% sum.swap_restrict
tff(fact_3584_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(B,A),A4: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),A4))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7311177749621191930dd_sum(B,real),aTP_Lamp_cq(fun(B,A),fun(B,real),F2)),A4)) ) ).

% norm_sum
tff(fact_3585_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,K4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),K4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),K4)) ) ) ).

% sum_mono
tff(fact_3586_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(nat),F2: fun(nat,A),G: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A4)
         => ( ! [X3: nat] :
                ( member(nat,aa(nat,nat,suc,X3),A4)
               => ( 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),A4) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),A4) ) ) ) ) ).

% sum_cong_Suc
tff(fact_3587_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F2: fun(B,A),I5: set(B),G: fun(B,A),I: B] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),F2),I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),I5) )
         => ( ! [I2: B] :
                ( member(B,I2,I5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,I2)),aa(B,A,G,I2)) )
           => ( member(B,I,I5)
             => ( aa(set(B),$o,finite_finite2(B),I5)
               => ( aa(B,A,F2,I) = aa(B,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_3588_sum__norm__le,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S: set(A),F2: fun(A,B),G: fun(A,real)] :
          ( ! [X3: A] :
              ( member(A,X3,S)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X3))),aa(A,real,G,X3)) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),G),S)) ) ) ).

% sum_norm_le
tff(fact_3589_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_3590_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)) ) ) ).

% sum_nonneg
tff(fact_3591_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A4: set(B)] :
          ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) != zero_zero(A) )
         => ~ ! [A5: B] :
                ( member(B,A5,A4)
               => ( aa(B,A,G,A5) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_3592_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => ( aa(A,B,G,X3) = zero_zero(B) ) )
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_3593_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
             => aa(real,$o,aa(real,fun(real,$o),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_3594_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N4: nat,F2: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N2)
               => aa(real,$o,aa(real,fun(real,$o),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_3595_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_cr(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A4) ) ) ) ).

% sum.inter_filter
tff(fact_3596_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I5: set(nat)] :
          ( summable(A,F2)
         => ( aa(set(nat),$o,finite_finite2(nat),I5)
           => ( ! [N2: nat] :
                  ( member(nat,N2,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2)) )
             => aa(A,$o,aa(A,fun(A,$o),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_3597_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A),Xa: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_cs(fun(nat,A),fun(A,fun(nat,A)),F2),Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xa))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_ct(fun(nat,A),fun(A,fun(nat,real)),F2),Z)) ) ) ) ).

% powser_insidea
tff(fact_3598_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S3: set(A),Ta: set(B),G: fun(B,C),I: fun(B,A),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( aa(set(B),$o,finite_finite2(B),Ta)
           => ( ! [X3: B] :
                  ( member(B,X3,Ta)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X3)) )
             => ( ! [X3: A] :
                    ( member(A,X3,S3)
                   => ? [Xa2: B] :
                        ( member(B,Xa2,Ta)
                        & ( aa(B,A,I,Xa2) = X3 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X3)),aa(B,C,G,Xa2)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),F2),S3)),aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_3599_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4) = zero_zero(B) )
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => ( aa(A,B,F2,X) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_3600_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
           => ( ? [X2: A] :
                  ( member(A,X2,A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X2)),aa(A,B,G,X2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_3601_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X1: A,Y1: A,X24: A,Y24: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X24)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y24) )
               => aa(A,$o,aa(A,fun(A,$o),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),X24),Y24)) )
           => ( aa(set(B),$o,finite_finite2(B),S)
             => ( ! [X3: B] :
                    ( member(B,X3,S)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X3)),aa(B,A,G,X3)) )
               => aa(A,$o,aa(A,fun(A,$o),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_3602_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,G,N2))
         => ( summable(A,F2)
           => ( summable(A,G)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),suminf(A,G)) ) ) ) ) ).

% suminf_le
tff(fact_3603_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X3: A] :
                  ( member(A,X3,A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4)) ) ) ) ) ).

% sum_strict_mono
tff(fact_3604_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = $ite(member(A,Xa,A4),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4))) ) ) ) ).

% sum.insert_if
tff(fact_3605_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S6: set(A),T6: set(B),S: set(A),I: fun(B,A),J: fun(A,B),T3: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S6)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( ! [A5: A] :
                  ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),S),S6))
                 => ( aa(B,A,I,aa(A,B,J,A5)) = A5 ) )
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),S),S6))
                   => member(B,aa(A,B,J,A5),aa(set(B),set(B),minus_minus(set(B),T3),T6)) )
               => ( ! [B5: B] :
                      ( member(B,B5,aa(set(B),set(B),minus_minus(set(B),T3),T6))
                     => ( aa(A,B,J,aa(B,A,I,B5)) = B5 ) )
                 => ( ! [B5: B] :
                        ( member(B,B5,aa(set(B),set(B),minus_minus(set(B),T3),T6))
                       => member(A,aa(B,A,I,B5),aa(set(A),set(A),minus_minus(set(A),S),S6)) )
                   => ( ! [A5: A] :
                          ( member(A,A5,S6)
                         => ( aa(A,C,G,A5) = zero_zero(C) ) )
                     => ( ! [B5: B] :
                            ( member(B,B5,T6)
                           => ( aa(B,C,H,B5) = zero_zero(C) ) )
                       => ( ! [A5: A] :
                              ( member(A,A5,S)
                             => ( aa(B,C,H,aa(A,B,J,A5)) = aa(A,C,G,A5) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),H),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_3606_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N4)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,N4)
               => ( aa(nat,A,F2,N2) = zero_zero(A) ) )
           => summable(A,F2) ) ) ) ).

% summable_finite
tff(fact_3607_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F2: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(A,fun(fun(nat,A),fun(nat,A)),C3),F2))
         => ( ( C3 != zero_zero(A) )
           => summable(A,F2) ) ) ) ).

% summable_mult_D
tff(fact_3608_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,power_power(A,zero_zero(A))) ) ).

% summable_zero_power
tff(fact_3609_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S3: set(A),F2: fun(A,B),B2: B,I: A] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ! [I2: A] :
                ( member(A,I2,S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S3) = B2 )
             => ( member(A,I,S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),B2) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_3610_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S3: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ! [I2: A] :
                ( member(A,I2,S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
           => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S3) = zero_zero(B) )
             => ( member(A,I,S3)
               => ( aa(A,B,F2,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_3611_sum_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(set(A),fun(A,B),aTP_Lamp_cu(fun(A,B),fun(set(A),fun(A,B)),G),B2)),A4) ) ) ) ).

% sum.inter_restrict
tff(fact_3612_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_cv(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_3613_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),E2: real] :
          ( summable(A,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ~ ! [N8: nat] :
                  ~ ! [M3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),M3)
                     => ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),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,M3,N3)))),E2) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3614_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_3615_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) ) ) ) ) ).

% sum_pos
tff(fact_3616_sum__bounded__above,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),F2: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),K4) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4))),K4)) ) ) ).

% sum_bounded_above
tff(fact_3617_sum__bounded__below,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),K4: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),K4),aa(A,B,F2,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4))),K4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)) ) ) ).

% sum_bounded_below
tff(fact_3618_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C2: set(A),A4: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),C2),A4))
                   => ( aa(A,B,G,A5) = zero_zero(B) ) )
               => ( ! [B5: A] :
                      ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),C2),B2))
                     => ( aa(A,B,H,B5) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B2) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C2) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_3619_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C2: set(A),A4: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),C2),A4))
                   => ( aa(A,B,G,A5) = zero_zero(B) ) )
               => ( ! [B5: A] :
                      ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),C2),B2))
                     => ( aa(A,B,H,B5) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),C2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),C2) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),B2) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_3620_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T3) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_3621_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_3622_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,H,X3) = zero_zero(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_3623_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X3) = zero_zero(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),S) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_3624_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B2: set(A),A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = 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),G),aa(set(A),set(A),minus_minus(set(A),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B2)) ) ) ) ) ).

% sum.subset_diff
tff(fact_3625_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A4: set(A),B2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = aa(B,B,minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B2)) ) ) ) ) ).

% sum_diff
tff(fact_3626_sum_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T3: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,finite_finite2(A),S)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,H,I2) = zero_zero(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),minus_minus(set(A),S),T3))
                   => ( aa(A,B,G,I2) = zero_zero(B) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3))
                     => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),T3) ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
tff(fact_3627_sum_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( 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),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) = 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),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B2)) ) ) ) ) ).

% sum.union_inter
tff(fact_3628_sum_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = 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),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),B2))) ) ) ) ).

% sum.Int_Diff
tff(fact_3629_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2))
           => ( ( suminf(A,F2) = zero_zero(A) )
            <=> ! [N: nat] : aa(nat,A,F2,N) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3630_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_nonneg
tff(fact_3631_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,N2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ).

% suminf_pos
tff(fact_3632_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_cw(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_0_powser
tff(fact_3633_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A)] : summable(A,aTP_Lamp_cx(fun(nat,A),fun(nat,A),F2)) ) ).

% summable_zero_power'
tff(fact_3634_sum_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_cy(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% sum.If_cases
tff(fact_3635_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( ? [N7: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
             => aa(real,$o,aa(real,fun(real,$o),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_cz(fun(nat,A),fun(nat,real),F2)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3636_summable__rabs__comparison__test,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
         => aa(real,$o,aa(real,fun(real,$o),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_da(fun(nat,real),fun(nat,real),F2)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3637_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B2: set(A),A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => ( ! [B5: A] :
                  ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),B2),A4))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,B5)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B2)) ) ) ) ) ).

% sum_mono2
tff(fact_3638_sum_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
                 => ( aa(A,B,G,X3) = zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = 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),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B2)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_3639_summable__rabs,axiom,
    ! [F2: fun(nat,real)] :
      ( summable(real,aTP_Lamp_da(fun(nat,real),fun(nat,real),F2))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),suminf(real,F2))),suminf(real,aTP_Lamp_da(fun(nat,real),fun(nat,real),F2))) ) ).

% summable_rabs
tff(fact_3640_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3641_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_3642_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A4: set(A),F2: fun(A,B),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = $ite(member(A,A3,A4),aa(B,B,minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(A,B,F2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)) ) ) ) ).

% sum_diff1
tff(fact_3643_sum__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A4: set(A),B2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(B,B,minus_minus(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),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ) ).

% sum_Un
tff(fact_3644_sum_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = 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),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),B2)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_3645_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = 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),minus_minus(set(A),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),minus_minus(set(A),B2),A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ).

% sum_Un2
tff(fact_3646_sum_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = 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),G),aa(set(A),set(A),minus_minus(set(A),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),minus_minus(set(A),B2),A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ) ).

% sum.union_diff2
tff(fact_3647_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N4)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,N4)
               => ( 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),N4) ) ) ) ) ).

% suminf_finite
tff(fact_3648_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2))
            <=> ? [I4: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I4)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3649_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),I: nat] :
          ( summable(A,F2)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),suminf(A,F2)) ) ) ) ) ).

% suminf_pos2
tff(fact_3650_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_db(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B3),C3)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B3,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C3),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),C3),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_3651_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_3652_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: 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,Na))) = 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),Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_3653_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = 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),Na))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_3654_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( 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,Na))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_3655_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(nat,A),Xa: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_dc(fun(nat,A),fun(A,fun(nat,A)),F2),Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xa))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_dc(fun(nat,A),fun(A,fun(nat,A)),F2),Z)) ) ) ) ).

% powser_inside
tff(fact_3656_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => summable(A,power_power(A,C3)) ) ) ).

% summable_geometric
tff(fact_3657_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => summable(A,power_power(A,Xa)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3658_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
         => ( suminf(A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),F2)) = aa(A,A,minus_minus(A,suminf(A,F2)),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3659_set__replicate__Suc,axiom,
    ! [A: $tType,Na: nat,Xa: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Na),Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_3660_set__replicate__conv__if,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      aa(list(A),set(A),set2(A),replicate(A,Na,Xa)) = $ite(Na = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_3661_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B2: set(A),A4: set(A),B3: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B2),A4))
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,B3))
               => ( ! [X3: A] :
                      ( member(A,X3,B2)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),B2)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_3662_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A4: set(A),F2: fun(A,B)] :
          ( member(A,I,A4)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
           => ( aa(set(A),$o,finite_finite2(A),A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)) ) ) ) ) ).

% member_le_sum
tff(fact_3663_sum__bounded__above__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere8940638589300402666id_add(B)
        & semiring_1(B) )
     => ! [A4: set(A),F2: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),K4) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A4))
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4))),K4)) ) ) ) ).

% sum_bounded_above_strict
tff(fact_3664_sum__bounded__above__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(B)
     => ! [A4: set(A),F2: fun(A,B),K4: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),divide_divide(B,K4,aa(nat,B,semiring_1_of_nat(B),aa(set(A),nat,finite_card(A),A4)))) )
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4)),K4) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_3665_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( 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,Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) = 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_de(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3666_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_df(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,aa(nat,nat,suc,Na))),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_3667_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xa: fun(A,B),A3: fun(A,B),B3: B,Delta: B] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xa,I2)) )
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Xa),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(A,B,A3,I2)),B3))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(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_dg(fun(A,B),fun(fun(A,B),fun(A,B)),Xa),A3)),I5)),B3))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3668_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_dh(fun(nat,A),fun(nat,real),F2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F2))),suminf(real,aTP_Lamp_dh(fun(nat,A),fun(nat,real),F2))) ) ) ).

% summable_norm
tff(fact_3669_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F2: fun(nat,A),A3: nat,B3: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or1337092689740270186AtMost(nat,A3,B3)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_di(fun(nat,A),fun(nat,fun(A,A)),F2),A3,B3,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3670_sum__norm__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S: set(A),F2: fun(A,B),K4: real] :
          ( ! [X3: A] :
              ( member(A,X3,S)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X3))),K4) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),S))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(set(A),nat,finite_card(A),S))),K4)) ) ) ).

% sum_norm_bound
tff(fact_3671_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),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),Na),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,Na))),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),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_3672_sum__div__partition,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [A4: set(A),F2: fun(A,B),B3: B] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4),B3) = 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),aa(B,fun(A,B),aTP_Lamp_dj(fun(A,B),fun(B,fun(A,B)),F2),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_dk(fun(A,B),fun(B,fun(A,$o)),F2),B3))))),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_dl(fun(A,B),fun(B,fun(A,$o)),F2),B3)))),B3)) ) ) ) ).

% sum_div_partition
tff(fact_3673_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_dc(fun(nat,A),fun(A,fun(nat,A)),F2),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_dc(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_dm(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3674_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_dc(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_dm(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),Z) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_dc(fun(nat,A),fun(A,fun(nat,A)),F2),Z))),aa(nat,A,F2,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3675_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
         => ( summable(A,F2)
           => ? [N8: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N3)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_dn(fun(nat,A),fun(nat,fun(nat,A)),F2),N3)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_3676_summable__power__series,axiom,
    ! [F2: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,I2))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Z)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_do(fun(nat,real),fun(real,fun(nat,real)),F2),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3677_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A3: fun(nat,A),M5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),R2),R0)
           => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A3,N2))),aa(nat,real,power_power(real,R0),N2))),M5)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_dp(real,fun(fun(nat,A),fun(nat,real)),R2),A3)) ) ) ) ) ).

% Abel_lemma
tff(fact_3678_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F2: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dq(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),aa(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),Na),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_3679_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_dr(fun(nat,A),fun(nat,A),F2)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,Na)),aa(nat,A,F2,M)) ) ) ) ).

% sum_telescope''
tff(fact_3680_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C3: real,N4: nat,F2: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),one_one(real))
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N2)
               => aa(real,$o,aa(real,fun(real,$o),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),C3),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2)))) )
           => summable(A,F2) ) ) ) ).

% summable_ratio_test
tff(fact_3681_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Na: nat] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)),one_one(A)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Na))) ) ).

% mask_eq_sum_exp
tff(fact_3682_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xa)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xa)),set_or1337092689740270186AtMost(nat,M,Na))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,Xa),M)),aa(nat,A,power_power(A,Xa),aa(nat,nat,suc,Na))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3683_even__sum__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),A4))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_ds(set(A),fun(fun(A,B),fun(A,$o)),A4),F2)))) ) ) ) ).

% even_sum_iff
tff(fact_3684_accp__subset__induct,axiom,
    ! [A: $tType,D3: fun(A,$o),R: fun(A,fun(A,$o)),Xa: A,P: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),D3),accp(A,R))
     => ( ! [X3: A,Z2: A] :
            ( aa(A,$o,D3,X3)
           => ( aa(A,$o,aa(A,fun(A,$o),R,Z2),X3)
             => aa(A,$o,D3,Z2) ) )
       => ( aa(A,$o,D3,Xa)
         => ( ! [X3: A] :
                ( aa(A,$o,D3,X3)
               => ( ! [Z3: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),R,Z3),X3)
                     => aa(A,$o,P,Z3) )
                 => aa(A,$o,P,X3) ) )
           => aa(A,$o,P,Xa) ) ) ) ) ).

% accp_subset_induct
tff(fact_3685_mask__eq__sum__exp__nat,axiom,
    ! [Na: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Na))) ).

% mask_eq_sum_exp_nat
tff(fact_3686_gauss__sum__nat,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dt(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_3687_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_du(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_3688_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,D2: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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_dv(A,fun(A,fun(nat,A)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) = 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),Na)),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),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D2))) ) ).

% double_arith_series
tff(fact_3689_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_3690_arith__series__nat,axiom,
    ! [A3: nat,D2: nat,Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_dw(nat,fun(nat,fun(nat,nat)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Na)),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),bit0(one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),D2))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_3691_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D2: A,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_dx(A,fun(A,fun(nat,A)),A3),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = 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),Na)),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),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),D2))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_3692_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: 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),Na)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_3693_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),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)),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_3694_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va3: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va3))) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va3))),
        $let(
          half: nat,
          half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va3)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va3)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_3695_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Na: 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)),Na)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Na)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_3696_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_dy(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)),aa(num,A,numeral_numeral(A),bit0(one2)))),gbinomial(A,R2,aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_3697_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xa: fun(A,B),Ya: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dz(set(A),fun(fun(A,B),fun(A,$o)),I5),Xa)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dz(set(A),fun(fun(A,B),fun(A,$o)),I5),Ya)))
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ea(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xa),Ya))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_3698_vebt__buildup_Opelims,axiom,
    ! [Xa: nat,Ya: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xa) = Ya )
     => ( aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),Xa)
       => ( ( ( Xa = zero_zero(nat) )
           => ( ( Ya = vEBT_Leaf($false,$false) )
             => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),zero_zero(nat)) ) )
         => ( ( ( Xa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Ya = vEBT_Leaf($false,$false) )
               => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va: nat] :
                  ( ( Xa = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
                 => ( ( Ya = $ite(
                          aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
                          $let(
                            half: nat,
                            half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_3699_Maclaurin__sin__expansion3,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),Xa)
            & ( sin(real,Xa) = 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_eb(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3700_Maclaurin__sin__expansion4,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ? [T5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),Xa)
          & ( sin(real,Xa) = 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_eb(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3701_Maclaurin__sin__expansion2,axiom,
    ! [Xa: real,Na: nat] :
    ? [T5: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa))
      & ( sin(real,Xa) = 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_eb(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_3702_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Na: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),Na)),aa(nat,A,power_power(A,Z),Na)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(nat,A,power_power(A,Z),aa(nat,nat,minus_minus(nat,Na),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_ed(A,fun(A,fun(nat,fun(nat,A))),H),Z),Na)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_3703_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,aa(A,set(A),set_ord_lessThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),K) ) ) ).

% lessThan_iff
tff(fact_3704_finite__lessThan,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% finite_lessThan
tff(fact_3705_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),Xa)),aa(A,set(A),set_ord_lessThan(A),Ya))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ).

% lessThan_subset_iff
tff(fact_3706_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3707_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_3708_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Na: A] :
          ( ! [X3: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Q,X3)),aa(A,nat,P,X3))
         => ( aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),P),aa(A,set(A),set_ord_lessThan(A),Na))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),Q),aa(A,set(A),set_ord_lessThan(A),Na))) = 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_ee(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),Na)) ) ) ) ).

% sum_diff_distrib
tff(fact_3709_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xa: A] : aa(A,set(A),set_ord_lessThan(A),Xa) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_3710_infinite__Iio,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A3: A] : ~ aa(set(A),$o,finite_finite2(A),aa(A,set(A),set_ord_lessThan(A),A3)) ) ).

% infinite_Iio
tff(fact_3711_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ef(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_3712_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [Na: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),Na) = bot_bot(set(A)) )
        <=> ( Na = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_3713_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Na: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M)),aa(A,set(A),set_ord_lessThan(A),Na))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3714_lessThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),K),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% lessThan_Suc
tff(fact_3715_lessThan__empty__iff,axiom,
    ! [Na: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),Na) = bot_bot(set(nat)) )
    <=> ( Na = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3716_sum__multicount__gen,axiom,
    ! [A: $tType,B: $tType,S3: set(A),Ta: set(B),R: fun(A,fun(B,$o)),K: fun(B,nat)] :
      ( aa(set(A),$o,finite_finite2(A),S3)
     => ( aa(set(B),$o,finite_finite2(B),Ta)
       => ( ! [X3: B] :
              ( member(B,X3,Ta)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ax(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),S3),R),X3))) = aa(B,nat,K,X3) ) )
         => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_eg(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Ta),R)),S3) = aa(set(B),nat,aa(fun(B,nat),fun(set(B),nat),groups7311177749621191930dd_sum(B,nat),K),Ta) ) ) ) ) ).

% sum_multicount_gen
tff(fact_3717_sum__subtractf__nat,axiom,
    ! [A: $tType,A4: set(A),G: fun(A,nat),F2: fun(A,nat)] :
      ( ! [X3: A] :
          ( member(A,X3,A4)
         => aa(nat,$o,aa(nat,fun(nat,$o),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_eh(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F2)),A4) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_3718_finite__nat__bounded,axiom,
    ! [S: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
     => ? [K2: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_lessThan(nat),K2)) ) ).

% finite_nat_bounded
tff(fact_3719_finite__nat__iff__bounded,axiom,
    ! [S: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_lessThan(nat),K3)) ) ).

% finite_nat_iff_bounded
tff(fact_3720_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X: A] :
            ( member(A,X,A4)
            & ( aa(A,nat,F2,X) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A4)
               => ( ( X != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_3721_sum__SucD,axiom,
    ! [A: $tType,F2: fun(A,nat),A4: set(A),Na: nat] :
      ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4) = aa(nat,nat,suc,Na) )
     => ? [X3: A] :
          ( member(A,X3,A4)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X3)) ) ) ).

% sum_SucD
tff(fact_3722_sum__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4) = one_one(nat) )
      <=> ? [X: A] :
            ( member(A,X,A4)
            & ( aa(A,nat,F2,X) = one_one(nat) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A4)
               => ( ( X != Xa3 )
                 => ( aa(A,nat,F2,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_3723_ivl__disj__int__one_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(4)
tff(fact_3724_sum__multicount,axiom,
    ! [A: $tType,B: $tType,S: set(A),T3: set(B),R: fun(A,fun(B,$o)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(B),$o,finite_finite2(B),T3)
       => ( ! [X3: B] :
              ( member(B,X3,T3)
             => ( aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ax(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),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,$o)),fun(A,nat),aTP_Lamp_eg(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),T3),R)),S) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(set(B),nat,finite_card(B),T3)) ) ) ) ) ).

% sum_multicount
tff(fact_3725_sum__diff__nat,axiom,
    ! [A: $tType,B2: set(A),A4: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B2)) ) ) ) ).

% sum_diff_nat
tff(fact_3726_sum__diff1__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),A4: set(A),A3: A] :
      aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = $ite(member(A,A3,A4),aa(nat,nat,minus_minus(nat,aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(A,nat,F2,A3)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)) ).

% sum_diff1_nat
tff(fact_3727_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [K: A,Xa: A] :
          aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),K),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))),bot_bot(set(A))) ) ).

% Iio_Int_singleton
tff(fact_3728_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xa: A] :
          ( summable(A,F2)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),N2))),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F2)),Xa) ) ) ) ).

% suminf_le_const
tff(fact_3729_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Na))) = 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_de(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3730_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_ei(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),aa(nat,A,F2,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3731_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_df(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,minus_minus(A,aa(nat,A,F2,M)),aa(nat,A,F2,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3732_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Xa: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,N2))
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),N2))),Xa)
           => summable(A,F2) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3733_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: 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)),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_de(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3734_sum__nth__roots,axiom,
    ! [Na: nat,C3: complex] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_ej(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aa(complex,fun(complex,$o),aTP_Lamp_ad(nat,fun(complex,fun(complex,$o)),Na),C3))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_3735_sum__Un__nat,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),B2))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ).

% sum_Un_nat
tff(fact_3736_finite__enumerate__initial__segment,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat,S3: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(A,set(A),set_ord_lessThan(A),S3))))
           => ( aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(A,set(A),set_ord_lessThan(A),S3))),Na) = aa(nat,A,infini527867602293511546merate(A,S),Na) ) ) ) ) ).

% finite_enumerate_initial_segment
tff(fact_3737_sum__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),Na)
     => ( aa(set(complex),complex,aa(fun(complex,complex),fun(set(complex),complex),groups7311177749621191930dd_sum(complex,complex),aTP_Lamp_ej(complex,complex)),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_au(nat,fun(complex,$o),Na))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_3738_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,M4)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),Na))),suminf(A,F2)) ) ) ) ).

% sum_less_suminf
tff(fact_3739_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Na: nat,F2: fun(nat,A),K4: A,K: nat] :
          ( ! [P4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P4),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,P4)),K4) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Na),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),K4)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3740_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),Na: nat,I: nat] :
          ( summable(A,F2)
         => ( ! [M4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F2,M4)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),I)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),aa(nat,A,F2,I))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(nat,set(nat),set_ord_lessThan(nat),Na))),suminf(A,F2)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3741_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xa: real,Na: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xa = zero_zero(real) )
         => ( ( Na != 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_ek(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xa),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3742_Maclaurin__lemma,axiom,
    ! [H: real,F2: fun(real,real),J: fun(nat,real),Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ? [B4: 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_el(real,fun(fun(nat,real),fun(nat,real)),H),J)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),B4),divide_divide(real,aa(nat,real,power_power(real,H),Na),semiring_char_0_fact(real,Na)))) ) ).

% Maclaurin_lemma
tff(fact_3743_Maclaurin__exp__le,axiom,
    ! [Xa: real,Na: nat] :
    ? [T5: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa))
      & ( exp(real,Xa) = 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_em(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ).

% Maclaurin_exp_le
tff(fact_3744_Maclaurin__sin__bound,axiom,
    ! [Xa: real,Na: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,sin(real,Xa)),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_eb(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Xa)),Na))) ).

% Maclaurin_sin_bound
tff(fact_3745_Sum__Icc__int,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),Na)
     => ( aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7311177749621191930dd_sum(int,int),aTP_Lamp_en(int,int)),set_or1337092689740270186AtMost(int,M,Na)) = divide_divide(int,aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Na),aa(int,int,aa(int,fun(int,int),plus_plus(int),Na),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,minus_minus(int,M),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_3746_sum__pos__lt__pair,axiom,
    ! [F2: fun(nat,real),K: nat] :
      ( summable(real,F2)
     => ( ! [D6: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D6)))),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)))),D6)),one_one(nat))))))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),F2),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(real,F2)) ) ) ).

% sum_pos_lt_pair
tff(fact_3747_Maclaurin__exp__lt,axiom,
    ! [Xa: real,Na: nat] :
      ( ( Xa != zero_zero(real) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T5))
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa))
            & ( exp(real,Xa) = 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_em(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3748_Maclaurin__cos__expansion2,axiom,
    ! [Xa: real,Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),Xa)
            & ( cos(real,Xa) = 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_eo(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3749_Maclaurin__minus__cos__expansion,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),zero_zero(real))
            & ( cos(real,Xa) = 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_eo(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3750_Maclaurin__cos__expansion,axiom,
    ! [Xa: real,Na: nat] :
    ? [T5: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa))
      & ( cos(real,Xa) = 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_eo(real,fun(nat,real),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T5),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),Na))),pi))),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_3751_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,Xa: fun(nat,fun(A,A)),Xaa: nat,Xb3: nat,Xc: A,Ya: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xa,Xaa,Xb3,Xc) = Ya )
     => ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),Xa),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xaa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xb3),Xc))))
       => ~ ( ( Ya = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xb3),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xb3,aa(A,A,aa(nat,fun(A,A),Xa,Xaa),Xc))) )
           => ~ aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),Xa),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xaa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xb3),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_3752_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3753_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A12: nat,A23: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A12),aa(A,product_prod(nat,A),product_Pair(nat,A,A23),A32))))
     => ( ! [F5: fun(nat,fun(A,A)),A5: nat,B5: nat,Acc2: A] :
            ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),F5),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A5),aa(A,product_prod(nat,A),product_Pair(nat,A,B5),Acc2))))
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B5),A5)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),one_one(nat))),B5),aa(A,A,aa(nat,fun(A,A),F5,A5),Acc2)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F5),A5),B5),Acc2) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A12),A23),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_3754_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F2: fun(nat,fun(A,A)),A3: nat,B3: nat,Acc: A] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)),F2),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A3),aa(A,product_prod(nat,A),product_Pair(nat,A,B3),Acc))))
     => ( set_fo6178422350223883121st_nat(A,F2,A3,B3,Acc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),B3),A3),Acc,set_fo6178422350223883121st_nat(A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B3,aa(A,A,aa(nat,fun(A,A),F2,A3),Acc))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_3755_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ep(nat,fun(nat,A),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_even_sum
tff(fact_3756_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_eq(nat,fun(nat,A),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) ) ) ) ).

% choose_odd_sum
tff(fact_3757_in__measure,axiom,
    ! [A: $tType,Xa: A,Ya: A,F2: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),measure(A,F2))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Ya)) ) ).

% in_measure
tff(fact_3758_in__finite__psubset,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A4),B2),finite_psubset(A))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
        & aa(set(A),$o,finite_finite2(A),B2) ) ) ).

% in_finite_psubset
tff(fact_3759_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,aa(A,set(A),set_ord_atMost(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),K) ) ) ).

% atMost_iff
tff(fact_3760_finite__atMost,axiom,
    ! [K: nat] : aa(set(nat),$o,finite_finite2(nat),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% finite_atMost
tff(fact_3761_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),Xa)),aa(A,set(A),set_ord_atMost(A),Ya))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ).

% atMost_subset_iff
tff(fact_3762_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),H),H2) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_3763_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_3764_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_3765_infinite__Iic,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A3: A] : ~ aa(set(A),$o,finite_finite2(A),aa(A,set(A),set_ord_atMost(A),A3)) ) ).

% infinite_Iic
tff(fact_3766_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_er(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_3767_atMost__atLeast0,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_atMost(nat),Na) = set_or1337092689740270186AtMost(nat,zero_zero(nat),Na) ).

% atMost_atLeast0
tff(fact_3768_atMost__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,K)),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% atMost_Suc
tff(fact_3769_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A,H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_or1337092689740270186AtMost(A,L2,H2)) ) ).

% not_Iic_le_Icc
tff(fact_3770_finite__nat__iff__bounded__le,axiom,
    ! [S: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
    <=> ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S),aa(nat,set(nat),set_ord_atMost(nat),K3)) ) ).

% finite_nat_iff_bounded_le
tff(fact_3771_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A3)),aa(A,set(A),set_ord_lessThan(A),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3772_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Na))) = 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_de(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Na))) ) ).

% sum.atMost_Suc_shift
tff(fact_3773_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_ei(fun(nat,A),fun(nat,A),F2)),aa(nat,set(nat),set_ord_atMost(nat),I)) = aa(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_3774_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Na: nat,D2: fun(nat,A)] :
          ( ! [X: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C3),X)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),D2),X)),aa(nat,set(nat),set_ord_atMost(nat),Na))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
             => ( aa(nat,A,C3,I4) = aa(nat,A,D2,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_3775_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: fun(nat,A),B2: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A3,N2))
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A3),aa(nat,set(nat),set_ord_atMost(nat),N2))),B2)
           => summable(A,A3) ) ) ) ).

% bounded_imp_summable
tff(fact_3776_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_3777_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_3778_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Na: nat] :
          ( ! [X: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C3),X)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
             => ( aa(nat,A,C3,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_3779_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C3: fun(nat,A),Na: nat,K: nat] :
          ( ! [W: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_et(fun(nat,A),fun(A,fun(nat,A)),C3),W)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => ( aa(nat,A,C3,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_3780_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Na)) = 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_de(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% sum.atMost_shift
tff(fact_3781_sum__choose__diagonal,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aa(nat,fun(nat,nat),aTP_Lamp_eu(nat,fun(nat,fun(nat,nat)),M),Na)),aa(nat,set(nat),set_ord_atMost(nat),M)) = binomial(aa(nat,nat,suc,Na),M) ) ) ).

% sum_choose_diagonal
tff(fact_3782_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C3,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(A,$o)),C3),Na))) ) ) ) ).

% polyfun_roots_finite
tff(fact_3783_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(A,$o)),C3),Na)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),Na)
              & ( aa(nat,A,C3,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_3784_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C3,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(A,$o)),C3),Na)))),Na) ) ) ) ).

% polyfun_roots_card
tff(fact_3785_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C3: fun(nat,A),A3: A,Na: nat] :
          ( ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),C3),A3)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A) )
         => ~ ! [B5: 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_ew(fun(nat,A),fun(A,fun(nat,A)),C3),Z3)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z3),A3)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),B5),Z3)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ) ).

% polyfun_linear_factor_root
tff(fact_3786_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Na: nat,Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xa)),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Xa),M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),power_power(A,Xa)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,Na),M)))) ) ) ) ).

% sum_power_shift
tff(fact_3787_atLeast1__atMost__eq__remove0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_atMost(nat),Na)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_3788_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),K: nat,Na: nat] :
          ( ( aa(nat,A,C3,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
           => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(A,$o)),C3),Na)))
              & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(nat,fun(A,$o),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(A,$o)),C3),Na)))),Na) ) ) ) ) ).

% polyfun_rootbound
tff(fact_3789_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A3: fun(nat,A),Na: nat,B3: fun(nat,A),Xa: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
             => ( aa(nat,A,A3,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),J2)
               => ( aa(nat,A,B3,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_ew(fun(nat,A),fun(A,fun(nat,A)),A3),Xa)),aa(nat,set(nat),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_ew(fun(nat,A),fun(A,fun(nat,A)),B3),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = 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_ey(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A3),B3),Xa)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ) ).

% polynomial_product
tff(fact_3790_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C3: fun(nat,A),Na: nat,K: A] :
          ( ! [X: A] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_es(fun(nat,A),fun(A,fun(nat,A)),C3),X)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = K
        <=> ( ( aa(nat,A,C3,zero_zero(nat)) = K )
            & ! [X: nat] :
                ( member(nat,X,set_or1337092689740270186AtMost(nat,one_one(nat),Na))
               => ( aa(nat,A,C3,X) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_3791_polynomial__product__nat,axiom,
    ! [M: nat,A3: fun(nat,nat),Na: nat,B3: fun(nat,nat),Xa: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),I2)
         => ( aa(nat,nat,A3,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),J2)
           => ( aa(nat,nat,B3,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_ez(fun(nat,nat),fun(nat,fun(nat,nat)),A3),Xa)),aa(nat,set(nat),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_ez(fun(nat,nat),fun(nat,fun(nat,nat)),B3),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = 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_fb(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A3),B3),Xa)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na))) ) ) ) ).

% polynomial_product_nat
tff(fact_3792_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),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_fc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),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_fd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_3793_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,Z: A,A3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( ( aa(nat,A,power_power(A,Z),Na) = A3 )
          <=> ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_fe(nat,fun(A,fun(A,fun(nat,A))),Na),Z),A3)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_3794_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( ( Na != one_one(nat) )
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ff(nat,fun(nat,A),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3795_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A3: fun(nat,A),Xa: A,Ya: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( aa(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_ew(fun(nat,A),fun(A,fun(nat,A)),A3),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),A3),Ya)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xa),Ya)),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_fh(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A3),Xa),Ya)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ) ) ).

% polyfun_diff_alt
tff(fact_3796_card__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bn(set(A),fun(nat,fun(list(A),$o)),A4),Na))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(set(A),nat,finite_card(A),A4))),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ) ).

% card_lists_length_le
tff(fact_3797_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_fi(nat,fun(nat,A),Na)),aa(nat,set(nat),set_ord_atMost(nat),Na)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3798_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C3: fun(nat,A),Na: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ? [M9: real] :
            ! [Z3: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M9),real_V7770717601297561774m_norm(A,Z3))
             => aa(real,$o,aa(real,fun(real,$o),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_cs(fun(nat,A),fun(A,fun(nat,A)),C3),Z3)),aa(nat,set(nat),set_ord_atMost(nat),Na)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,Na)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_3799_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,A3: fun(nat,A),Xa: A,Ya: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Na)
         => ( aa(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_ew(fun(nat,A),fun(A,fun(nat,A)),A3),Xa)),aa(nat,set(nat),set_ord_atMost(nat),Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),A3),Ya)),aa(nat,set(nat),set_ord_atMost(nat),Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xa),Ya)),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_fk(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Na),A3),Xa),Ya)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ) ) ).

% polyfun_diff
tff(fact_3800_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => sums(A,aTP_Lamp_fl(A,fun(nat,A),Z),divide_divide(A,one_one(A),aa(nat,A,power_power(A,aa(A,A,minus_minus(A,one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_3801_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [M4: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,M4)),aa(nat,A,X4,N2)) )
         => topological_monoseq(A,X4) ) ) ).

% monoI1
tff(fact_3802_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [M4: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N2)),aa(nat,A,X4,M4)) )
         => topological_monoseq(A,X4) ) ) ).

% monoI2
tff(fact_3803_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( topological_monoseq(A,X4)
        <=> ( ! [M2: nat,N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,M2)),aa(nat,A,X4,N)) )
            | ! [M2: nat,N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N)),aa(nat,A,X4,M2)) ) ) ) ) ).

% monoseq_def
tff(fact_3804_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => sums(A,aTP_Lamp_bx(nat,A),zero_zero(A)) ) ).

% sums_zero
tff(fact_3805_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A),Xa: A] :
          ( sums(A,aTP_Lamp_cw(fun(nat,A),fun(nat,A),A3),Xa)
        <=> ( aa(nat,A,A3,zero_zero(nat)) = Xa ) ) ) ).

% powser_sums_zero_iff
tff(fact_3806_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_3807_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(nat,A),G: fun(nat,A),S3: A,Ta: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N2)),aa(nat,A,G,N2))
         => ( sums(A,F2,S3)
           => ( sums(A,G,Ta)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S3),Ta) ) ) ) ) ).

% sums_le
tff(fact_3808_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_by(nat,fun(fun(nat,A),fun(nat,A)),I),F2),aa(nat,A,F2,I)) ) ).

% sums_single
tff(fact_3809_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F2: fun(nat,A),D2: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fm(A,fun(fun(nat,A),fun(nat,A)),C3),F2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),C3))
          <=> sums(A,F2,D2) ) ) ) ).

% sums_mult2_iff
tff(fact_3810_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C3: A,F2: fun(nat,A),D2: A] :
          ( ( C3 != zero_zero(A) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fn(A,fun(fun(nat,A),fun(nat,A)),C3),F2),aa(A,A,aa(A,fun(A,A),times_times(A),C3),D2))
          <=> sums(A,F2,D2) ) ) ) ).

% sums_mult_iff
tff(fact_3811_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F2: fun(nat,A),A3: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(A,fun(fun(nat,A),fun(nat,A)),C3),F2),A3)
         => ( ( C3 != zero_zero(A) )
           => sums(A,F2,divide_divide(A,A3,C3)) ) ) ) ).

% sums_mult_D
tff(fact_3812_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S3: A] :
          ( ( aa(nat,A,F2,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),F2),S3)
           => sums(A,F2,S3) ) ) ) ).

% sums_Suc_imp
tff(fact_3813_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F2: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_fo(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_3814_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),S3: A] :
          ( sums(A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),F2),S3)
        <=> sums(A,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3815_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Na: nat,F2: fun(nat,A),S3: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
             => ( aa(nat,A,F2,I2) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_fp(nat,fun(fun(nat,A),fun(nat,A)),Na),F2),S3)
          <=> sums(A,F2,S3) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3816_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A4: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),A4)
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cf(set(nat),fun(fun(nat,A),fun(nat,A)),A4),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),A4)) ) ) ).

% sums_If_finite_set
tff(fact_3817_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),aa(fun(nat,$o),set(nat),collect(nat),P))
         => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ce(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F2),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),aa(fun(nat,$o),set(nat),collect(nat),P))) ) ) ).

% sums_If_finite
tff(fact_3818_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N4: set(nat),F2: fun(nat,A)] :
          ( aa(set(nat),$o,finite_finite2(nat),N4)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,N4)
               => ( 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),N4)) ) ) ) ).

% sums_finite
tff(fact_3819_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_fq(nat,fun(A,fun(nat,A)),M),Z),aa(nat,A,power_power(A,Z),M)) ) ).

% powser_sums_if
tff(fact_3820_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A)] : sums(A,aTP_Lamp_cw(fun(nat,A),fun(nat,A),A3),aa(nat,A,A3,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3821_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S: A,A4: set(nat),S6: A,F2: fun(nat,A)] :
          ( sums(A,G,S)
         => ( aa(set(nat),$o,finite_finite2(nat),A4)
           => ( ( S6 = 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_fr(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F2)),A4)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_fs(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A4),F2),S6) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_3822_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,C3)),one_one(real))
         => sums(A,power_power(A,C3),divide_divide(A,one_one(A),aa(A,A,minus_minus(A,one_one(A)),C3))) ) ) ).

% geometric_sums
tff(fact_3823_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N2)),aa(nat,A,X4,aa(nat,nat,suc,N2)))
         => topological_monoseq(A,X4) ) ) ).

% mono_SucI1
tff(fact_3824_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,aa(nat,nat,suc,N2))),aa(nat,A,X4,N2))
         => topological_monoseq(A,X4) ) ) ).

% mono_SucI2
tff(fact_3825_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( topological_monoseq(A,X4)
        <=> ( ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N)),aa(nat,A,X4,aa(nat,nat,suc,N)))
            | ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,aa(nat,nat,suc,N))),aa(nat,A,X4,N)) ) ) ) ).

% monoseq_Suc
tff(fact_3826_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Ya: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_fu(A,fun(A,fun(nat,A)),Xa),Ya),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xa)),sin(A,Ya))) ) ).

% sin_x_sin_y
tff(fact_3827_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Ya: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_fw(A,fun(A,fun(nat,A)),Xa),Ya),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya))) ) ).

% sums_cos_x_plus_y
tff(fact_3828_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,Ya: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_fy(A,fun(A,fun(nat,A)),Xa),Ya),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xa)),cos(A,Ya))) ) ).

% cos_x_cos_y
tff(fact_3829_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C3: fun(nat,A),Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),C3),Xa))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_ga(fun(nat,A),fun(A,fun(nat,A)),C3),Xa),suminf(A,aa(A,fun(nat,A),aTP_Lamp_fz(fun(nat,A),fun(A,fun(nat,A)),C3),Xa))) ) ) ).

% diffs_equiv
tff(fact_3830_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real] : aa(A,A,real_V8093663219630862766scaleR(A,A3),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_3831_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,Xa: A,B3: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,B3),Xa) )
        <=> ( ( A3 = B3 )
            | ( Xa = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_3832_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,Xa: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(real) )
            | ( Xa = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_3833_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),Xa) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_3834_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,A3: real,B3: real] :
          ( ( Xa != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,B3),Xa) )
           => ( A3 = B3 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_3835_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B3: real,A3: real,C3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),C3)),aa(A,A,real_V8093663219630862766scaleR(A,B3),C3)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3836_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B3),Xa)) ) ) ) ).

% scaleR_right_mono
tff(fact_3837_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_3838_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_3839_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B3))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) )
            & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_3840_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B3: A,A3: A,C3: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C3),B3)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_3841_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xa: A,Ya: A,A3: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Ya)) ) ) ) ).

% scaleR_left_mono
tff(fact_3842_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,U: real,V2: real,A3: A] :
          ( ( Xa = aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V2)),A3) )
        <=> $ite(V2 = zero_zero(real),Xa = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V2),Xa) = aa(A,A,real_V8093663219630862766scaleR(A,U),A3)) ) ) ).

% eq_vector_fraction_iff
tff(fact_3843_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V2: real,A3: A,Xa: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,U,V2)),A3) = Xa )
        <=> $ite(V2 = zero_zero(real),Xa = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A3) = aa(A,A,real_V8093663219630862766scaleR(A,V2),Xa)) ) ) ).

% vector_fraction_eq_iff
tff(fact_3844_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E2: A,C3: A,B3: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),E2)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B3),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A3),B3)),E2)),C3)),D2) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3845_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E2: A,C3: A,B3: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),E2)),C3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B3),E2)),D2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,B3),A3)),E2)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3846_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_3847_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)),zero_zero(A))
        <=> ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_3848_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_3849_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3850_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3851_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3852_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)) ) ) ).

% split_scaleR_pos_le
tff(fact_3853_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xa: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_3854_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,C3: A,D2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),D2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),C3)),aa(A,A,real_V8093663219630862766scaleR(A,B3),D2)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3855_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,Xa: A,Ya: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,B3),Ya)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3856_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xa: A,A3: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)),Xa) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3857_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3858_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B3: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),B3) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3859_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),B3) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3860_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B3: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3861_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B3: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_3862_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),B3) ) ) ) ).

% pos_less_divideR_eq
tff(fact_3863_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B3: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3)),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),B3) ) ) ) ).

% neg_divideR_less_eq
tff(fact_3864_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_3865_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: real,Xa: A] :
          ( ( A3 != zero_zero(real) )
         => ( ( Xa != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xa)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A3)),aa(A,A,inverse_inverse(A),Xa)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_3866_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B3: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3867_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3868_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B3: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3869_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3870_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B3: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3871_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3872_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,B3: A,A3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3))),A3)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3873_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,A3: A,B3: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C3)),B3)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C3),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_3874_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,K4: real,C3: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),K4)
         => ( ! [X3: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X3)),K4)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),X3)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_gc(A,fun(fun(nat,A),fun(nat,A)),Xa),C3)) ) ) ) ).

% termdiff_converges
tff(fact_3875_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : sums(A,aTP_Lamp_gd(A,fun(nat,A),Xa),cosh(A,Xa)) ) ).

% cosh_converges
tff(fact_3876_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xa: A] : sums(A,aTP_Lamp_ge(A,fun(nat,A),Xa),sinh(A,Xa)) ) ).

% sinh_converges
tff(fact_3877_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),Na) = semiri8178284476397505188at_aux(A,aTP_Lamp_gf(A,A),Na,zero_zero(A)) ) ).

% of_nat_code
tff(fact_3878_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: fun(int,fun(int,$o))] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,A0),A12))
     => ( ! [I2: int,J2: int] :
            ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,I2),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A12) ) ) ).

% upto.pinduct
tff(fact_3879_arctan__def,axiom,
    ! [Ya: real] : arctan(Ya) = the(real,aTP_Lamp_gg(real,fun(real,$o),Ya)) ).

% arctan_def
tff(fact_3880_arcsin__def,axiom,
    ! [Ya: real] : aa(real,real,arcsin,Ya) = the(real,aTP_Lamp_gh(real,fun(real,$o),Ya)) ).

% arcsin_def
tff(fact_3881_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Na: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Na),I) = semiri8178284476397505188at_aux(A,Inc,Na,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_3882_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_3883_ln__neg__is__const,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),zero_zero(real))
     => ( aa(real,real,ln_ln(real),Xa) = the(real,aTP_Lamp_gi(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_3884_the__elem__def,axiom,
    ! [A: $tType,X4: set(A)] : the_elem(A,X4) = the(A,aTP_Lamp_gj(set(A),fun(A,$o),X4)) ).

% the_elem_def
tff(fact_3885_arccos__def,axiom,
    ! [Ya: real] : aa(real,real,arccos,Ya) = the(real,aTP_Lamp_gk(real,fun(real,$o),Ya)) ).

% arccos_def
tff(fact_3886_pi__half,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))) = the(real,aTP_Lamp_gl(real,$o)) ).

% pi_half
tff(fact_3887_pi__def,axiom,
    pi = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),the(real,aTP_Lamp_gl(real,$o))) ).

% pi_def
tff(fact_3888_divmod__step__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),aa(fun(A,fun(A,product_prod(A,A))),fun(product_prod(A,A),product_prod(A,A)),product_case_prod(A,A,product_prod(A,A)),aTP_Lamp_gm(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_3889_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X4)
     => ( aa(set(A),$o,finite_finite2(A),X4)
       => ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),count_list(A,Xs)),X4) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_3890_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Ya: $o] :
      ( ( vEBT_VEBT_minNull(Xa)
      <=> (Ya) )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ( (Ya)
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xa = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Ya)
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xa = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Ya)
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                    ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy) )
                   => ( (Ya)
                     => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy)) ) )
               => ~ ! [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                      ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc) )
                     => ( ~ (Ya)
                       => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_3891_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] :
          comm_s3205402744901411588hammer(A,A3,Na) = $ite(Na = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_gn(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_3892_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_3893_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% pochhammer_Suc0
tff(fact_3894_count__notin,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,nat,count_list(A,Xs),Xa) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_3895_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,Na)) ) ) ).

% pochhammer_pos
tff(fact_3896_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,M) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( comm_s3205402744901411588hammer(A,A3,Na) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_3897_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Na: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,Na) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
           => ( comm_s3205402744901411588hammer(A,A3,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_3898_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xa: A,Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xa,Na)) ) ) ).

% pochhammer_nonneg
tff(fact_3899_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Na: nat] :
          comm_s3205402744901411588hammer(A,zero_zero(A),Na) = $ite(Na = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% pochhammer_0_left
tff(fact_3900_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),Xa)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% count_le_length
tff(fact_3901_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Na: nat,K: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_3902_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Na: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_3903_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,Na: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,Na) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K3),Na)
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_3904_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Na)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_3905_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Na: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( comm_s3205402744901411588hammer(A,Z,Na) = 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,minus_minus(nat,Na),M))) ) ) ) ).

% pochhammer_product
tff(fact_3906_floor__real__def,axiom,
    ! [Xa: real] : archim6421214686448440834_floor(real,Xa) = the(int,aTP_Lamp_go(real,fun(int,$o),Xa)) ).

% floor_real_def
tff(fact_3907_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xa)
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ! [Uv2: $o] :
              ( ( Xa = vEBT_Leaf($true,(Uv2)) )
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xa = vEBT_Leaf((Uu2),$true) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz: product_prod(nat,nat),Va2: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va2,Vb,Vc)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_3908_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xa)
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xa)
       => ( ( ( Xa = vEBT_Leaf($false,$false) )
           => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_3909_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_gp(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_3910_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),aa(fun(int,fun(int,product_prod(int,int))),fun(product_prod(int,int),product_prod(int,int)),product_case_prod(int,int,product_prod(int,int)),aTP_Lamp_gq(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_3911_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Na: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Na))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gr(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),bit0(one2))),Na)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3912_divmod__nat__if,axiom,
    ! [M: nat,Na: nat] :
      divmod_nat(M,Na) = $ite(
        ( ( Na = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ),
        aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),M),
        aa(product_prod(nat,nat),product_prod(nat,nat),aa(fun(nat,fun(nat,product_prod(nat,nat))),fun(product_prod(nat,nat),product_prod(nat,nat)),product_case_prod(nat,nat,product_prod(nat,nat)),aTP_Lamp_gs(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),Na),Na)) ) ).

% divmod_nat_if
tff(fact_3913_floor__rat__def,axiom,
    ! [Xa: rat] : archim6421214686448440834_floor(rat,Xa) = the(int,aTP_Lamp_gt(rat,fun(int,$o),Xa)) ).

% floor_rat_def
tff(fact_3914_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F3: set(A),I5: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),F3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_gu(set(A),fun(fun(A,B),fun(A,$o)),I5),F2))),F3)
           => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_3915_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4) = zero_zero(B) )
          <=> ? [X: A] :
                ( member(A,X,A4)
                & ( aa(A,B,F2,X) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_3916_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_3917_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_3918_dvd__prod__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A4: set(A),A3: A,B3: B,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => ( ( B3 = aa(A,B,F2,A3) )
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),B3),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)) ) ) ) ) ).

% dvd_prod_eqI
tff(fact_3919_dvd__prodI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [A4: set(A),A3: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)) ) ) ) ).

% dvd_prodI
tff(fact_3920_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_3921_sum_Oeq__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1027152243600224163dd_sum(A,B,P3,I5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),P3),I5) ) ) ) ).

% sum.eq_sum
tff(fact_3922_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gv(A,fun(fun(A,B),fun(A,B)),A3),B3)),S) = $ite(member(A,A3,S),aa(A,B,B3,A3),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_3923_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aTP_Lamp_gw(A,fun(fun(A,B),fun(A,B)),A3),B3)),S) = $ite(member(A,A3,S),aa(A,B,B3,A3),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_3924_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)) ) ) ) ) ).

% prod.insert
tff(fact_3925_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dz(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( groups1027152243600224163dd_sum(A,B,P3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),I5)) = $ite(member(A,I,I5),groups1027152243600224163dd_sum(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P3,I)),groups1027152243600224163dd_sum(A,B,P3,I5))) ) ) ) ).

% sum.insert'
tff(fact_3926_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na)))) ) ).

% prod.cl_ivl_Suc
tff(fact_3927_abs__rat__def,axiom,
    ! [A3: rat] :
      aa(rat,rat,abs_abs(rat),A3) = $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),A3),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A3),A3) ).

% abs_rat_def
tff(fact_3928_sgn__rat__def,axiom,
    ! [A3: rat] :
      sgn_sgn(rat,A3) = $ite(
        A3 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A3),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_3929_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F2: fun(B,A),A4: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F2),A4))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7121269368397514597t_prod(B,real),aTP_Lamp_gx(fun(B,A),fun(B,real),F2)),A4)) ) ).

% norm_prod_le
tff(fact_3930_less__eq__rat__def,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xa),Ya)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xa),Ya)
        | ( Xa = Ya ) ) ) ).

% less_eq_rat_def
tff(fact_3931_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
     => ~ ! [S4: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),S4)
           => ! [T5: rat] :
                ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),T5)
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S4),T5) ) ) ) ) ).

% obtain_pos_sum
tff(fact_3932_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),B2: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),$o,finite_finite2(B),B2)
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_gy(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B2),G),R)),A4) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_ha(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A4),G),R)),B2) ) ) ) ) ).

% prod.swap_restrict
tff(fact_3933_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_hb(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_3934_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)) ) ) ).

% prod_nonneg
tff(fact_3935_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)) ) ) ).

% prod_mono
tff(fact_3936_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(A,B,F2,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)) ) ) ).

% prod_pos
tff(fact_3937_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)) ) ) ).

% prod_ge_1
tff(fact_3938_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ? [X2: A] :
                ( member(A,X2,A4)
                & ( aa(A,B,F2,X2) = zero_zero(B) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_3939_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,$o),fun(A,B),aTP_Lamp_hc(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A4) ) ) ) ).

% prod.inter_filter
tff(fact_3940_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_hd(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hf(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ).

% sum.triangle_reindex_eq
tff(fact_3941_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,X3))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_3942_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X1: A,Y1: A,X24: A,Y24: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X1),X24)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y1),Y24) )
               => aa(A,$o,aa(A,fun(A,$o),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),X24),Y24)) )
           => ( aa(set(B),$o,finite_finite2(B),S)
             => ( ! [X3: B] :
                    ( member(B,X3,S)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X3)),aa(B,A,G,X3)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S)) ) ) ) ) ) ).

% prod.related
tff(fact_3943_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = $ite(member(A,Xa,A4),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4))) ) ) ) ).

% prod.insert_if
tff(fact_3944_prod__dvd__prod__subset2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [B2: set(A),A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(A,B,F2,A5)),aa(A,B,G,A5)) )
             => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_3945_prod__dvd__prod__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B2: set(A),A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B2)) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_3946_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S6: set(A),T6: set(B),S: set(A),I: fun(B,A),J: fun(A,B),T3: set(B),G: fun(A,C),H: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S6)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( ! [A5: A] :
                  ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),S),S6))
                 => ( aa(B,A,I,aa(A,B,J,A5)) = A5 ) )
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),S),S6))
                   => member(B,aa(A,B,J,A5),aa(set(B),set(B),minus_minus(set(B),T3),T6)) )
               => ( ! [B5: B] :
                      ( member(B,B5,aa(set(B),set(B),minus_minus(set(B),T3),T6))
                     => ( aa(A,B,J,aa(B,A,I,B5)) = B5 ) )
                 => ( ! [B5: B] :
                        ( member(B,B5,aa(set(B),set(B),minus_minus(set(B),T3),T6))
                       => member(A,aa(B,A,I,B5),aa(set(A),set(A),minus_minus(set(A),S),S6)) )
                   => ( ! [A5: A] :
                          ( member(A,A5,S6)
                         => ( aa(A,C,G,A5) = one_one(C) ) )
                     => ( ! [B5: B] :
                            ( member(B,B5,T6)
                           => ( aa(B,C,H,B5) = one_one(C) ) )
                       => ( ! [A5: A] :
                              ( member(A,A5,S)
                             => ( aa(B,C,H,aa(A,B,J,A5)) = aa(A,C,G,A5) ) )
                         => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),G),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),H),T3) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3947_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hg(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_3948_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7311177749621191930dd_sum(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_hh(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_hf(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% sum.triangle_reindex
tff(fact_3949_prod_Ointer__restrict,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(set(A),fun(A,B),aTP_Lamp_hi(fun(A,B),fun(set(A),fun(A,B)),G),B2)),A4) ) ) ) ).

% prod.inter_restrict
tff(fact_3950_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_hj(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3951_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( exp(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),F2),I5)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aTP_Lamp_hk(fun(A,B),fun(A,B),F2)),I5) ) ) ) ).

% exp_sum
tff(fact_3952_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,I2)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3953_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X3) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S) = groups1027152243600224163dd_sum(A,B,G,T3) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_3954_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X3) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T3) = groups1027152243600224163dd_sum(A,B,G,S) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_3955_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,H,I2) = zero_zero(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S) = groups1027152243600224163dd_sum(A,B,H,T3) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_3956_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X3) = zero_zero(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T3) = groups1027152243600224163dd_sum(A,B,H,S) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_3957_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(A,B,F2,I2)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),I5)) ) ) ) ) ).

% less_1_prod
tff(fact_3958_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B2: set(A),A4: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3959_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X3) = one_one(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3960_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,H,X3) = one_one(B) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
               => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3961_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3962_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,G,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),T3) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3963_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C2: set(A),A4: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),C2),A4))
                   => ( aa(A,B,G,A5) = one_one(B) ) )
               => ( ! [B5: A] :
                      ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),C2),B2))
                     => ( aa(A,B,H,B5) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C2) )
                   => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B2) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3964_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C2: set(A),A4: set(A),B2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),C2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
             => ( ! [A5: A] :
                    ( member(A,A5,aa(set(A),set(A),minus_minus(set(A),C2),A4))
                   => ( aa(A,B,G,A5) = one_one(B) ) )
               => ( ! [B5: A] :
                      ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),C2),B2))
                     => ( aa(A,B,H,B5) = one_one(B) ) )
                 => ( ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),B2) )
                  <=> ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),C2) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),C2) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3965_prod_Ounion__inter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ).

% prod.union_inter
tff(fact_3966_prod_OInt__Diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),B2))) ) ) ) ).

% prod.Int_Diff
tff(fact_3967_prod_Omono__neutral__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T3: set(A),S: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( aa(set(A),$o,finite_finite2(A),S)
           => ( ! [I2: A] :
                  ( member(A,I2,aa(set(A),set(A),minus_minus(set(A),T3),S))
                 => ( aa(A,B,H,I2) = one_one(B) ) )
             => ( ! [I2: A] :
                    ( member(A,I2,aa(set(A),set(A),minus_minus(set(A),S),T3))
                   => ( aa(A,B,G,I2) = one_one(B) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3))
                     => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
                 => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),S) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),T3) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_3968_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3969_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3970_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Na))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3971_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dz(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_dz(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hg(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_3972_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_hb(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_hb(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_3973_prod_OIf__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),P: fun(A,$o),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_hl(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),P),H),G)),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),aa(fun(A,$o),set(A),collect(A),P))))) ) ) ) ).

% prod.If_cases
tff(fact_3974_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3975_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))),aa(nat,A,G,aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hm(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Na))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3976_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Na))) ) ).

% prod.atMost_Suc_shift
tff(fact_3977_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3978_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [I2: A] :
                ( member(A,I2,A4)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,I2)),aa(A,B,G,I2)) ) )
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)) ) ) ) ) ).

% prod_mono_strict
tff(fact_3979_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4))
          <=> ? [X: A] :
                ( member(A,X,A4)
                & aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,F2,X)) ) ) ) ) ).

% even_prod_iff
tff(fact_3980_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_3981_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),Xa: A,G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xa)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_3982_prod_Ounion__inter__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
                 => ( aa(A,B,G,X3) = one_one(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_3983_prod_Ounion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),B2)) ) ) ) ) ) ).

% prod.union_disjoint
tff(fact_3984_prod_Ounion__diff2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),B2: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),A4),B2))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),minus_minus(set(A),B2),A4)))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ) ).

% prod.union_diff2
tff(fact_3985_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A),P3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),P3)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3986_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_hn(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B3),C3)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C3),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),C3),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_3987_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I2))),one_one(real)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W2,I2))),one_one(real)) )
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Z),I5)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),W2),I5)))),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aa(fun(A,B),fun(A,real),aTP_Lamp_ho(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W2)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_3988_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hm(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ) ).

% prod.atMost_shift
tff(fact_3989_fact__eq__fact__times,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Na)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_dt(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Na),M))) ) ) ).

% fact_eq_fact_times
tff(fact_3990_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B2: set(A),A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => ( ! [B5: A] :
                  ( member(A,B5,aa(set(A),set(A),minus_minus(set(A),B2),A4))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F2,B5)) )
             => ( ! [A5: A] :
                    ( member(A,A5,A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,A5)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B2)) ) ) ) ) ) ).

% prod_mono2
tff(fact_3991_prod__le__power,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),F2: fun(A,B),Na: B,K: nat] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F2,I2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),Na) ) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),K)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),Na)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(nat,B,power_power(B,Na),K)) ) ) ) ) ).

% prod_le_power
tff(fact_3992_prod__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( field(B)
     => ! [A4: set(A),B2: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ! [X3: A] :
                  ( member(A,X3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
                 => ( aa(A,B,F2,X3) != zero_zero(B) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = divide_divide(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),B2)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ) ) ).

% prod_Un
tff(fact_3993_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A4: set(A),F2: fun(A,B),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = $ite(member(A,A3,A4),divide_divide(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4),aa(A,B,F2,A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F2),A4)) ) ) ) ) ).

% prod_diff1
tff(fact_3994_prod__gen__delta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),A3: A,B3: fun(A,B),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_hp(A,fun(fun(A,B),fun(B,fun(A,B))),A3),B3),C3)),S) = $ite(member(A,A3,S),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,A3)),aa(nat,B,power_power(B,C3),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),S)),one_one(nat)))),aa(nat,B,power_power(B,C3),aa(set(A),nat,finite_card(A),S))) ) ) ) ).

% prod_gen_delta
tff(fact_3995_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hq(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod
tff(fact_3996_fact__div__fact,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( divide_divide(nat,semiring_char_0_fact(nat,M),semiring_char_0_fact(nat,Na)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_dt(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_3997_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hr(A,fun(nat,fun(nat,A)),A3),Na)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3998_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F2: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_gu(set(A),fun(fun(A,B),fun(A,$o)),I5),F2)))
         => ( groups1027152243600224163dd_sum(A,B,F2,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F2,I5)),aa(A,B,F2,I)),groups1027152243600224163dd_sum(A,B,F2,I5)) ) ) ) ).

% sum_diff1'
tff(fact_3999_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),P3)
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_hs(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P3)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_ht(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,P3),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_4000_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : gbinomial(A,A3,aa(nat,nat,suc,K)) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hu(A,fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_4001_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(nat,A,semiring_1_of_nat(A),Na) = $ite(Na = zero_zero(nat),zero_zero(A),aa(product_prod(nat,nat),A,aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),aTP_Lamp_hv(nat,fun(nat,A))),divmod_nat(Na,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_4002_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A4: set(A),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(set(A),nat,finite_card(A),A4))
       => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_hw(set(A),fun(nat,fun(list(A),$o)),A4),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_dt(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A4)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A4))) ) ) ) ).

% card_lists_distinct_length_eq
tff(fact_4003_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A4: set(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(set(A),nat,finite_card(A),A4))
     => ( aa(set(list(A)),nat,finite_card(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(set(A),fun(list(A),$o),aTP_Lamp_hx(nat,fun(set(A),fun(list(A),$o)),K),A4))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_dt(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A4)),K)),one_one(nat)),aa(set(A),nat,finite_card(A),A4))) ) ) ).

% card_lists_distinct_length_eq'
tff(fact_4004_normalize__negative,axiom,
    ! [Q5: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Q5),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q5)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),P3)),aa(int,int,uminus_uminus(int),Q5))) ) ) ).

% normalize_negative
tff(fact_4005_prod__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A4) = one_one(nat) )
      <=> ! [X: A] :
            ( member(A,X,A4)
           => ( aa(A,nat,F2,X) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_4006_prod__pos__nat__iff,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F2),A4))
      <=> ! [X: A] :
            ( member(A,X,A4)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(A,nat,F2,X)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_4007_distinct__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( distinct(A,list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_4008_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(set(list(A)),$o,finite_finite2(list(A)),aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_hw(set(A),fun(nat,fun(list(A),$o)),A4),Na))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4009_finite__distinct__list,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A4 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_4010_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I: nat,J: nat] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => ( ( aa(nat,A,nth(A,Xs),I) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_4011_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
         => ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( I4 != J3 )
               => ( aa(nat,A,nth(A,Xs),I4) != aa(nat,A,nth(A,Xs),J3) ) ) ) ) ) ).

% distinct_conv_nth
tff(fact_4012_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( distinct(A,Xs)
     => ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
       => ? [X3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs))
            & ( aa(nat,A,nth(A,Xs),X3) = Xa )
            & ! [Y2: nat] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Y2),aa(list(A),nat,size_size(list(A)),Xs))
                  & ( aa(nat,A,nth(A,Xs),Y2) = Xa ) )
               => ( Y2 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4013_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P3: int,Q5: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q5) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q5) ) ).

% normalize_denom_pos
tff(fact_4014_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F2: fun(A,real)] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,I2)) )
       => ( aa(real,real,ln_ln(real),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),F2),I5)) = aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7311177749621191930dd_sum(A,real),aTP_Lamp_hy(fun(A,real),fun(A,real),F2)),I5) ) ) ) ).

% ln_prod
tff(fact_4015_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A3: A,I: nat] :
      ( distinct(A,Xs)
     => ( ~ member(A,A3,aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),I)),bot_bot(set(A)))))
       => distinct(A,list_update(A,Xs,I,A3)) ) ) ).

% distinct_list_update
tff(fact_4016_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_hd(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ia(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),Na)) ) ).

% prod.triangle_reindex_eq
tff(fact_4017_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Na: nat] : aa(set(product_prod(nat,nat)),A,aa(fun(product_prod(nat,nat),A),fun(set(product_prod(nat,nat)),A),groups7121269368397514597t_prod(product_prod(nat,nat),A),aa(fun(nat,fun(nat,A)),fun(product_prod(nat,nat),A),product_case_prod(nat,nat,A),G)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_hh(nat,fun(nat,fun(nat,$o)),Na)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ia(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),Na)) ) ).

% prod.triangle_reindex
tff(fact_4018_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),Na: nat,Xa: A] :
      ( distinct(A,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,Na,Xa)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(nat,A,nth(A,Xs),Na)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_4019_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,$o)),B2: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A4),B2)
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A4))),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),B2))) ) ).

% Collect_case_prod_mono
tff(fact_4020_finite__enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(set(A),nat,finite_card(A),S))
           => ( aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,Na)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_ib(set(A),fun(nat,fun(A,$o)),S),Na)) ) ) ) ) ).

% finite_enumerate_Suc''
tff(fact_4021_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_4022_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_4023_Least__eq__0,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_4024_LeastI2,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A3)
         => ( ! [X3: A] :
                ( aa(A,$o,P,X3)
               => aa(A,$o,Q,X3) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2
tff(fact_4025_LeastI__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% LeastI_ex
tff(fact_4026_LeastI2__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
         => ( ! [X3: A] :
                ( aa(A,$o,P,X3)
               => aa(A,$o,Q,X3) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_ex
tff(fact_4027_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% LeastI
tff(fact_4028_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
         => ( ! [A5: A] :
                ( aa(A,$o,P,A5)
               => ( ! [B10: A] :
                      ( aa(A,$o,P,B10)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B10) )
                 => aa(A,$o,Q,A5) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_4029_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A3)
         => ( ! [A5: A] :
                ( aa(A,$o,P,A5)
               => ( ! [B10: A] :
                      ( aa(A,$o,P,B10)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),B10) )
                 => aa(A,$o,Q,A5) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder
tff(fact_4030_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xa: A] :
          ( aa(A,$o,P,Xa)
         => ( ! [Y: A] :
                ( aa(A,$o,P,Y)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) )
           => ( ord_Least(A,P) = Xa ) ) ) ) ).

% Least_equality
tff(fact_4031_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xa: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xa)
         => ( ! [Y: A] :
                ( aa(A,$o,P,Y)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) )
           => ( ! [X3: A] :
                  ( aa(A,$o,P,X3)
                 => ( ! [Y2: A] :
                        ( aa(A,$o,P,Y2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y2) )
                   => aa(A,$o,Q,X3) ) )
             => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ) ).

% LeastI2_order
tff(fact_4032_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Z: A] :
          ( ? [X2: A] :
              ( aa(A,$o,P,X2)
              & ! [Y: A] :
                  ( aa(A,$o,P,Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y) )
              & ! [Y: A] :
                  ( ( aa(A,$o,P,Y)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Ya2) ) )
                 => ( Y = X2 ) ) )
         => ( aa(A,$o,P,Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),Z) ) ) ) ).

% Least1_le
tff(fact_4033_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] :
          ( ? [X2: A] :
              ( aa(A,$o,P,X2)
              & ! [Y: A] :
                  ( aa(A,$o,P,Y)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y) )
              & ! [Y: A] :
                  ( ( aa(A,$o,P,Y)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Ya2) ) )
                 => ( Y = X2 ) ) )
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% Least1I
tff(fact_4034_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),K) ) ) ).

% Least_le
tff(fact_4035_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),ord_Least(A,P))
         => ~ aa(A,$o,P,K) ) ) ).

% not_less_Least
tff(fact_4036_Least__Suc2,axiom,
    ! [P: fun(nat,$o),Na: nat,Q: fun(nat,$o),M: nat] :
      ( aa(nat,$o,P,Na)
     => ( aa(nat,$o,Q,M)
       => ( ~ aa(nat,$o,P,zero_zero(nat))
         => ( ! [K2: nat] :
                ( aa(nat,$o,P,aa(nat,nat,suc,K2))
              <=> aa(nat,$o,Q,K2) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_4037_Least__Suc,axiom,
    ! [P: fun(nat,$o),Na: nat] :
      ( aa(nat,$o,P,Na)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_ic(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_4038_size__list__estimation,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Ya: nat,F2: fun(A,nat)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ya),aa(A,nat,F2,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ya),size_list(A,F2,Xs)) ) ) ).

% size_list_estimation
tff(fact_4039_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F2: fun(A,nat),G: fun(A,nat)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,X3)),aa(A,nat,G,X3)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),size_list(A,F2,Xs)),size_list(A,G,Xs)) ) ).

% size_list_pointwise
tff(fact_4040_size__list__estimation_H,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Ya: nat,F2: fun(A,nat)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ya),aa(A,nat,F2,Xa))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ya),size_list(A,F2,Xs)) ) ) ).

% size_list_estimation'
tff(fact_4041_enumerate__0,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A)] : aa(nat,A,infini527867602293511546merate(A,S),zero_zero(nat)) = ord_Least(A,aTP_Lamp_id(set(A),fun(A,$o),S)) ) ).

% enumerate_0
tff(fact_4042_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_ie(set(A),fun(set(A),$o)))) ).

% finite_psubset_def
tff(fact_4043_enumerate__Suc_H_H,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => ( aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,Na)) = ord_Least(A,aa(nat,fun(A,$o),aTP_Lamp_ib(set(A),fun(nat,fun(A,$o)),S),Na)) ) ) ) ).

% enumerate_Suc''
tff(fact_4044_enumerate__Suc,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A),Na: nat] : aa(nat,A,infini527867602293511546merate(A,S),aa(nat,nat,suc,Na)) = aa(nat,A,infini527867602293511546merate(A,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),ord_Least(A,aTP_Lamp_id(set(A),fun(A,$o),S))),bot_bot(set(A))))),Na) ) ).

% enumerate_Suc
tff(fact_4045_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_if(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_4046_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = aa(fun(product_prod(int,int),$o),set(product_prod(int,int)),collect(product_prod(int,int)),aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_ig(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_4047_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_4048_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys2)) = binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_4049_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or7035219750837199246ssThan(nat,L,U)) ).

% finite_atLeastLessThan
tff(fact_4050_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_4051_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( set_or7035219750837199246ssThan(A,A3,B3) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_4052_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,Na: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,Na))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),I)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Na) ) ) ) ) ).

% ivl_subset
tff(fact_4053_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_4054_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B3) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4055_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% infinite_Ico_iff
tff(fact_4056_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,Na: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),Na)
         => ( aa(set(A),set(A),minus_minus(set(A),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,Na)) = set_or7035219750837199246ssThan(A,Na,M) ) ) ) ).

% ivl_diff
tff(fact_4057_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_4058_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          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,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),zero_zero(A),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,Na))),aa(nat,A,G,Na))) ) ).

% sum.op_ivl_Suc
tff(fact_4059_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% prod.op_ivl_Suc
tff(fact_4060_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
           => ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C3,D2) )
            <=> ( ( A3 = C3 )
                & ( B3 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4061_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C3,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
             => ( A3 = C3 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4062_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C3,D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),C3),D2)
             => ( B3 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4063_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B3)),set_or7035219750837199246ssThan(A,C3,D2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4064_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(set(A),$o,finite_finite2(A),set_or7035219750837199246ssThan(A,A3,B3)) ) ) ).

% infinite_Ico
tff(fact_4065_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_4066_ex__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ? [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Na)
          & aa(nat,$o,P,M2) )
    <=> ? [X: nat] :
          ( member(nat,X,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less_eq
tff(fact_4067_all__nat__less__eq,axiom,
    ! [Na: nat,P: fun(nat,$o)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),Na)
         => aa(nat,$o,P,M2) )
    <=> ! [X: nat] :
          ( member(nat,X,set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less_eq
tff(fact_4068_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_4069_lessThan__atLeast0,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_lessThan(nat),Na) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ).

% lessThan_atLeast0
tff(fact_4070_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_4071_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys2: list(A)] :
      ( member(list(A),Zs,shuffles(A,Xs,Ys2))
     => ( aa(list(A),set(A),set2(A),Zs) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ) ) ).

% set_shuffles
tff(fact_4072_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A3: A,C3: A,B3: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A3 = C3 )
         => ( ( B3 = D2 )
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D2)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),set_or7035219750837199246ssThan(A,A3,B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),set_or7035219750837199246ssThan(A,C3,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4073_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A3: A,C3: A,B3: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A3 = C3 )
         => ( ( B3 = D2 )
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),D2)
                   => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) ) )
             => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),set_or7035219750837199246ssThan(A,A3,B3)) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),set_or7035219750837199246ssThan(A,C3,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4074_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_4075_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_4076_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),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,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,Na,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_4077_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,P3: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P3)
           => ( aa(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,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,Na,P3)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4078_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_4079_ivl__disj__int__one_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(2)
tff(fact_4080_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,P3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),P3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,Na,P3))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,P3)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_4081_atLeast0__lessThan__Suc,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Na),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ).

% atLeast0_lessThan_Suc
tff(fact_4082_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N4),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => aa(set(nat),$o,finite_finite2(nat),N4) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_4083_subset__card__intvl__is__intvl,axiom,
    ! [A4: set(nat),K: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),A4),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A4))))
     => ( A4 = set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(set(nat),nat,finite_card(nat),A4))) ) ) ).

% subset_card_intvl_is_intvl
tff(fact_4084_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),set_or7035219750837199246ssThan(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4085_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A3,B3)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4086_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_4087_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_4088_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: 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,Na))) = 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),Na))),aa(nat,A,G,Na)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_4089_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) = 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),Na))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4090_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: nat,B3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,A3,B3))),aa(nat,A,G,B3)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_4091_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] : set_or7035219750837199246ssThan(A,A3,B3) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4092_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(nat,A,G,Na)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4093_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),Na))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4094_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B3))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,A3,B3))),aa(nat,A,G,B3)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_4095_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Na)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))) ) ) ) ).

% sum.last_plus
tff(fact_4096_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Na)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))) ) ) ) ).

% prod.last_plus
tff(fact_4097_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( set_or7035219750837199246ssThan(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),set_or7035219750837199246ssThan(nat,I,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_4098_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Na: nat,F2: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_df(fun(nat,A),fun(nat,A),F2)),set_or7035219750837199246ssThan(nat,M,Na)) = aa(A,A,minus_minus(A,aa(nat,A,F2,Na)),aa(nat,A,F2,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4099_atLeastLessThanSuc,axiom,
    ! [M: nat,Na: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Na),set_or7035219750837199246ssThan(nat,M,Na)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_4100_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_ih(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ij(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% sum.nested_swap
tff(fact_4101_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ik(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_im(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% prod.nested_swap
tff(fact_4102_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( distinct(A,Xs)
     => ( distinct(A,Ys2)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
         => ( member(list(A),Zs,shuffles(A,Xs,Ys2))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_4103_prod__Suc__Suc__fact,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Na)) = semiring_char_0_fact(nat,Na) ).

% prod_Suc_Suc_fact
tff(fact_4104_prod__Suc__fact,axiom,
    ! [Na: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) = semiring_char_0_fact(nat,Na) ).

% prod_Suc_fact
tff(fact_4105_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set(nat),Na: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N4),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N4)),Na) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_4106_card__sum__le__nat__sum,axiom,
    ! [S: set(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dt(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_dt(nat,nat)),S)) ).

% card_sum_le_nat_sum
tff(fact_4107_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_4108_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),zero_zero(A),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,Na))),aa(nat,A,G,Na))) ) ).

% sum.head_if
tff(fact_4109_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] :
          aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na))),aa(nat,A,G,Na))) ) ).

% prod.head_if
tff(fact_4110_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% fact_prod_Suc
tff(fact_4111_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,Na: nat] : comm_s3205402744901411588hammer(A,A3,Na) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hq(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% pochhammer_prod
tff(fact_4112_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Na: nat] : semiring_char_0_fact(A,Na) = aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),minus_minus(nat,Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% fact_prod_rev
tff(fact_4113_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A)] :
          ( summable(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N6: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M2)
                 => ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),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,M2,N)))),E3) ) ) ) ) ).

% summable_Cauchy
tff(fact_4114_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A,K: nat] :
          ( sums(A,F2,S3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_in(fun(nat,A),fun(nat,fun(nat,A)),F2),K),S3) ) ) ) ).

% sums_group
tff(fact_4115_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aTP_Lamp_io(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ).

% take_bit_sum
tff(fact_4116_atLeast1__lessThan__eq__remove0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Na) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_lessThan(nat),Na)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_4117_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( semiring_char_0_fact(A,Na) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,Na),K),Na)))),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Na),K))) ) ) ) ).

% fact_split
tff(fact_4118_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
         => ( aa(nat,A,semiring_1_of_nat(A),binomial(Na,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_ip(nat,fun(nat,fun(nat,A)),K),Na)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4119_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : gbinomial(A,A3,K) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_iq(A,fun(nat,fun(nat,A)),A3),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4120_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,A3,K)),semiring_char_0_fact(A,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ir(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_4121_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),gbinomial(A,A3,K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ir(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_4122_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : gbinomial(A,A3,K) = divide_divide(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hu(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_4123_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F2: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_is(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_4124_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Na: nat,A3: fun(nat,A),B3: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,I2)),aa(nat,A,A3,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B3,J2)),aa(nat,A,B3,I2)) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),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_it(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_4125_Chebyshev__sum__upper__nat,axiom,
    ! [Na: nat,A3: fun(nat,nat),B3: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A3,I2)),aa(nat,nat,A3,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),J2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J2),Na)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B3,J2)),aa(nat,nat,B3,I2)) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),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_iu(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A3),B3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_4126_divmod__step__integer__def,axiom,
    ! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_iv(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_4127_sgn__integer__code,axiom,
    ! [K: code_integer] :
      sgn_sgn(code_integer,K) = $ite(
        K = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_4128_less__eq__integer__code_I1_J,axiom,
    aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_eq_integer_code(1)
tff(fact_4129_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_4130_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(divide_divide(int,K,aa(num,int,numeral_numeral(int),bit0(one2))))),
            $ite(modulo_modulo(int,K,aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_4131_card__Pow,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow2(A,A4)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(A),nat,finite_card(A),A4)) ) ) ).

% card_Pow
tff(fact_4132_bezw__0,axiom,
    ! [Xa: nat] : bezw(Xa,zero_zero(nat)) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_4133_prod__decode__aux_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Ya )
     => ( Ya = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xa),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),aa(nat,nat,minus_minus(nat,Xa),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xa),aa(nat,nat,minus_minus(nat,Xaa),aa(nat,nat,suc,Xa)))) ) ) ).

% prod_decode_aux.elims
tff(fact_4134_Pow__empty,axiom,
    ! [A: $tType] : pow2(A,bot_bot(set(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_empty
tff(fact_4135_Pow__singleton__iff,axiom,
    ! [A: $tType,X4: set(A),Y3: set(A)] :
      ( ( pow2(A,X4) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Y3),bot_bot(set(set(A)))) )
    <=> ( ( X4 = bot_bot(set(A)) )
        & ( Y3 = bot_bot(set(A)) ) ) ) ).

% Pow_singleton_iff
tff(fact_4136_Pow__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( member(set(A),A4,pow2(A,B2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ).

% Pow_iff
tff(fact_4137_PowI,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => member(set(A),A4,pow2(A,B2)) ) ).

% PowI
tff(fact_4138_Pow__Int__eq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) = 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,A4)),pow2(A,B2)) ).

% Pow_Int_eq
tff(fact_4139_finite__Pow__iff,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),pow2(A,A4))
    <=> aa(set(A),$o,finite_finite2(A),A4) ) ).

% finite_Pow_iff
tff(fact_4140_abs__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,abs_abs(code_integer),K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_4141_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_4142_Un__Pow__subset,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),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,A4)),pow2(A,B2))),pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))) ).

% Un_Pow_subset
tff(fact_4143_less__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),code_integer_of_int(Xa)),code_integer_of_int(Xb3))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Xb3) ) ).

% less_integer.abs_eq
tff(fact_4144_PowD,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( member(set(A),A4,pow2(A,B2))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ).

% PowD
tff(fact_4145_Pow__mono,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pow2(A,A4)),pow2(A,B2)) ) ).

% Pow_mono
tff(fact_4146_Pow__top,axiom,
    ! [A: $tType,A4: set(A)] : member(set(A),A4,pow2(A,A4)) ).

% Pow_top
tff(fact_4147_Pow__not__empty,axiom,
    ! [A: $tType,A4: set(A)] : pow2(A,A4) != bot_bot(set(set(A))) ).

% Pow_not_empty
tff(fact_4148_Pow__bottom,axiom,
    ! [A: $tType,B2: set(A)] : member(set(A),bot_bot(set(A)),pow2(A,B2)) ).

% Pow_bottom
tff(fact_4149_Pow__def,axiom,
    ! [A: $tType,A4: set(A)] : pow2(A,A4) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ac(set(A),fun(set(A),$o),A4)) ).

% Pow_def
tff(fact_4150_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,Xb3: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),code_integer_of_int(Xa)),code_integer_of_int(Xb3))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xb3) ) ).

% less_eq_integer.abs_eq
tff(fact_4151_binomial__def,axiom,
    ! [Na: nat,K: nat] : binomial(Na,K) = aa(set(set(nat)),nat,finite_card(set(nat)),aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),aa(nat,fun(set(nat),$o),aTP_Lamp_iw(nat,fun(nat,fun(set(nat),$o)),Na),K))) ).

% binomial_def
tff(fact_4152_prod__decode__aux_Osimps,axiom,
    ! [K: nat,M: nat] :
      nat_prod_decode_aux(K,M) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),K),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M),aa(nat,nat,minus_minus(nat,K),M)),nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,K)))) ).

% prod_decode_aux.simps
tff(fact_4153_prod__decode__aux_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(Xa,Xaa) = Ya )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Xaa))
       => ~ ( ( Ya = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xaa),Xa),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xaa),aa(nat,nat,minus_minus(nat,Xa),Xaa)),nat_prod_decode_aux(aa(nat,nat,suc,Xa),aa(nat,nat,minus_minus(nat,Xaa),aa(nat,nat,suc,Xa)))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Xaa)) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_4154_card__partition,axiom,
    ! [A: $tType,C2: set(set(A)),K: nat] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),C2)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2))
       => ( ! [C5: set(A)] :
              ( member(set(A),C5,C2)
             => ( aa(set(A),nat,finite_card(A),C5) = K ) )
         => ( ! [C1: set(A),C22: set(A)] :
                ( member(set(A),C1,C2)
               => ( member(set(A),C22,C2)
                 => ( ( 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)),C2)) = aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2)) ) ) ) ) ) ).

% card_partition
tff(fact_4155_finite__enumerate,axiom,
    ! [S: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
     => ? [R3: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R3,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(nat),nat,finite_card(nat),S)))
          & ! [N3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N3),aa(set(nat),nat,finite_card(nat),S))
             => member(nat,aa(nat,nat,R3,N3),S) ) ) ) ).

% finite_enumerate
tff(fact_4156_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),product_Pair(code_integer,$o,zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,$o)),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o)),aTP_Lamp_ix(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% bit_cut_integer_code
tff(fact_4157_Sup__bot__conv_I2_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),A4) )
        <=> ! [X: A] :
              ( member(A,X,A4)
             => ( X = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(2)
tff(fact_4158_Sup__bot__conv_I1_J,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A4) = bot_bot(A) )
        <=> ! [X: A] :
              ( member(A,X,A4)
             => ( X = bot_bot(A) ) ) ) ) ).

% Sup_bot_conv(1)
tff(fact_4159_Sup__empty,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).

% Sup_empty
tff(fact_4160_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Xa,Ya)) = Ya ) ) ) ).

% Sup_atLeastAtMost
tff(fact_4161_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Ya,Xa)) = Xa ) ) ) ).

% cSup_atLeastAtMost
tff(fact_4162_cSup__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ) ).

% cSup_singleton
tff(fact_4163_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Ya,Xa)) = Xa ) ) ) ).

% cSup_atLeastLessThan
tff(fact_4164_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Xa,Ya)) = Ya ) ) ) ).

% Sup_atLeastLessThan
tff(fact_4165_Sup__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ).

% Sup_insert
tff(fact_4166_finite__Union,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A4)
     => ( ! [M9: set(A)] :
            ( member(set(A),M9,A4)
           => aa(set(A),$o,finite_finite2(A),M9) )
       => aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)) ) ) ).

% finite_Union
tff(fact_4167_Union__Un__distrib,axiom,
    ! [A: $tType,A4: set(set(A)),B2: set(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))),sup_sup(set(set(A))),A4),B2)) = 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)),A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B2)) ).

% Union_Un_distrib
tff(fact_4168_subset__Pow__Union,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),A4),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4))) ).

% subset_Pow_Union
tff(fact_4169_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,S: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S))
        <=> ? [X: A] :
              ( member(A,X,S)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X) ) ) ) ).

% less_Sup_iff
tff(fact_4170_Union__insert,axiom,
    ! [A: $tType,A3: set(A),B2: set(set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A3),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B2)) ).

% Union_insert
tff(fact_4171_Union__empty,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),bot_bot(set(set(A)))) = bot_bot(set(A)) ).

% Union_empty
tff(fact_4172_Union__empty__conv,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4) = bot_bot(set(A)) )
    <=> ! [X: set(A)] :
          ( member(set(A),X,A4)
         => ( X = bot_bot(set(A)) ) ) ) ).

% Union_empty_conv
tff(fact_4173_empty__Union__conv,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4) )
    <=> ! [X: set(A)] :
          ( member(set(A),X,A4)
         => ( X = bot_bot(set(A)) ) ) ) ).

% empty_Union_conv
tff(fact_4174_Union__subsetI,axiom,
    ! [A: $tType,A4: set(set(A)),B2: set(set(A))] :
      ( ! [X3: set(A)] :
          ( member(set(A),X3,A4)
         => ? [Y2: set(A)] :
              ( member(set(A),Y2,B2)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Y2) ) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B2)) ) ).

% Union_subsetI
tff(fact_4175_Union__upper,axiom,
    ! [A: $tType,B2: set(A),A4: set(set(A))] :
      ( member(set(A),B2,A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)) ) ).

% Union_upper
tff(fact_4176_Union__least,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(A)] :
      ( ! [X6: set(A)] :
          ( member(set(A),X6,A4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),C2) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)),C2) ) ).

% Union_least
tff(fact_4177_Union__mono,axiom,
    ! [A: $tType,A4: set(set(A)),B2: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),A4),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B2)) ) ).

% Union_mono
tff(fact_4178_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A4: set(A),V2: A] :
          ( member(A,U,A4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),U)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ).

% Sup_upper2
tff(fact_4179_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),B3)
        <=> ! [X: A] :
              ( member(A,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B3) ) ) ) ).

% Sup_le_iff
tff(fact_4180_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,A4: set(A)] :
          ( member(A,Xa,A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ).

% Sup_upper
tff(fact_4181_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),Z: A] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),Z) ) ) ).

% Sup_least
tff(fact_4182_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( ! [A5: A] :
              ( member(A,A5,A4)
             => ? [X2: A] :
                  ( member(A,X2,B2)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),X2) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ).

% Sup_mono
tff(fact_4183_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),Xa: A] :
          ( ! [Y: A] :
              ( member(A,Y,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) )
         => ( ! [Y: A] :
                ( ! [Z3: A] :
                    ( member(A,Z3,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z3),Y) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A4) = Xa ) ) ) ) ).

% Sup_eqI
tff(fact_4184_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X4: set(A)] :
          ( member(A,Z,X4)
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X4) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_4185_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X4: set(A),A3: A] :
          ( ! [X3: A] :
              ( member(A,X3,X4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A3) )
         => ( ! [Y: A] :
                ( ! [X2: A] :
                    ( member(A,X2,X4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Y) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X4) = A3 ) ) ) ) ).

% cSup_eq
tff(fact_4186_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xa: A,A4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),A4))
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xa)
             => ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X) ) ) ) ) ).

% le_Sup_iff
tff(fact_4187_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),Z: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X4)),Z) ) ) ) ).

% cSup_least
tff(fact_4188_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),A3: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A3) )
           => ( ! [Y: A] :
                  ( ! [X2: A] :
                      ( member(A,X2,X4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Y) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X4) = A3 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_4189_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V3) )
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ).

% less_eq_Sup
tff(fact_4190_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( member(A,Xa,X4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),X4)) ) ) ) ).

% le_cSup_finite
tff(fact_4191_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ).

% Sup_subset_mono
tff(fact_4192_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Ya: A,X4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),aa(set(A),A,complete_Sup_Sup(A),X4))
         => ( ( X4 != bot_bot(set(A)) )
           => ~ ! [X3: A] :
                  ( member(A,X3,X4)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),X3) ) ) ) ) ).

% less_cSupE
tff(fact_4193_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Z: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X4))
           => ? [X3: A] :
                ( member(A,X3,X4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),X3) ) ) ) ) ).

% less_cSupD
tff(fact_4194_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Xa: A,A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( member(A,Xa,X4)
           => ( ! [X3: A] :
                  ( member(A,X3,X4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),A3) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X4)),A3) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_4195_Sup__inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B2: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),B2)),A3) = bot_bot(A) )
        <=> ! [X: A] :
              ( member(A,X,B2)
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),A3) = bot_bot(A) ) ) ) ) ).

% Sup_inf_eq_bot_iff
tff(fact_4196_Sup__union__distrib,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ).

% Sup_union_distrib
tff(fact_4197_insert__partition,axiom,
    ! [A: $tType,Xa: set(A),F3: set(set(A))] :
      ( ~ member(set(A),Xa,F3)
     => ( ! [X3: set(A)] :
            ( member(set(A),X3,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Xa),F3))
           => ! [Xa4: set(A)] :
                ( member(set(A),Xa4,aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Xa),F3))
               => ( ( X3 != Xa4 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Xa),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F3)) = bot_bot(set(A)) ) ) ) ).

% insert_partition
tff(fact_4198_Union__disjoint,axiom,
    ! [A: $tType,C2: set(set(A)),A4: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2)),A4) = bot_bot(set(A)) )
    <=> ! [X: set(A)] :
          ( member(set(A),X,C2)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X),A4) = bot_bot(set(A)) ) ) ) ).

% Union_disjoint
tff(fact_4199_Union__Int__subset,axiom,
    ! [A: $tType,A4: set(set(A)),B2: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A4),B2))),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)),A4)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B2))) ).

% Union_Int_subset
tff(fact_4200_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2)) ).

% card_Union_le_sum_card
tff(fact_4201_finite__UnionD,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4))
     => aa(set(set(A)),$o,finite_finite2(set(A)),A4) ) ).

% finite_UnionD
tff(fact_4202_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X4)),A3)
            <=> ! [X: A] :
                  ( member(A,X,X4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_4203_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S))),A3) ) ) ) ).

% cSup_abs_le
tff(fact_4204_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2))) ) ).

% Sup_inter_less_eq
tff(fact_4205_finite__Sup__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] :
                  ( member(A,X3,A4)
                 => ( member(A,Y,A4)
                   => member(A,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y),A4) ) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),A4),A4) ) ) ) ) ).

% finite_Sup_in
tff(fact_4206_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X3: set(A)] :
          ( member(set(A),X3,U2)
         => aa(set(A),$o,finite_finite2(A),X3) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),U2)) ) ).

% card_Union_le_sum_card_weak
tff(fact_4207_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A4) = aa(set(A),A,complete_Sup_Sup(A),A4) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_4208_cSup__eq__Sup__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X4) = aa(set(A),A,lattic5882676163264333800up_fin(A),X4) ) ) ) ) ).

% cSup_eq_Sup_fin
tff(fact_4209_finite__subset__Union,axiom,
    ! [A: $tType,A4: set(A),B12: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12))
       => ~ ! [F6: set(set(A))] :
              ( aa(set(set(A)),$o,finite_finite2(set(A)),F6)
             => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),F6),B12)
               => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F6)) ) ) ) ) ).

% finite_subset_Union
tff(fact_4210_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X3),L))),E2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(set(A),A,complete_Sup_Sup(A),S)),L))),E2) ) ) ) ).

% cSup_asclose
tff(fact_4211_Sup__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),S)) = $ite(S = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,complete_Sup_Sup(A),S))) ) ) ) ).

% Sup_insert_finite
tff(fact_4212_dvd__partition,axiom,
    ! [A: $tType,C2: set(set(A)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2))
     => ( ! [X3: set(A)] :
            ( member(set(A),X3,C2)
           => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),X3)) )
       => ( ! [X3: set(A)] :
              ( member(set(A),X3,C2)
             => ! [Xa4: set(A)] :
                  ( member(set(A),Xa4,C2)
                 => ( ( X3 != Xa4 )
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2))) ) ) ) ).

% dvd_partition
tff(fact_4213_ccpo__Sup__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xa: A] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ) ).

% ccpo_Sup_singleton
tff(fact_4214_ccSup__empty,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = bot_bot(A) ) ) ).

% ccSup_empty
tff(fact_4215_card__UNION,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),A4)
     => ( ! [X3: set(A)] :
            ( member(set(A),X3,A4)
           => aa(set(A),$o,finite_finite2(A),X3) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A4)) = aa(int,nat,nat2,aa(set(set(set(A))),int,aa(fun(set(set(A)),int),fun(set(set(set(A))),int),groups7311177749621191930dd_sum(set(set(A)),int),aTP_Lamp_iy(set(set(A)),int)),aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_iz(set(set(A)),fun(set(set(A)),$o),A4)))) ) ) ) ).

% card_UNION
tff(fact_4216_Sup__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ).

% Sup_finite_insert
tff(fact_4217_finite__Inter,axiom,
    ! [A: $tType,M5: set(set(A))] :
      ( ? [X2: set(A)] :
          ( member(set(A),X2,M5)
          & aa(set(A),$o,finite_finite2(A),X2) )
     => aa(set(A),$o,finite_finite2(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),M5)) ) ).

% finite_Inter
tff(fact_4218_Sup__nat__empty,axiom,
    aa(set(nat),nat,complete_Sup_Sup(nat),bot_bot(set(nat))) = zero_zero(nat) ).

% Sup_nat_empty
tff(fact_4219_Inf__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),A4) = bot_bot(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X)
             => ? [Xa3: A] :
                  ( member(A,Xa3,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa3),X) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_4220_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Ya,Xa)) = Ya ) ) ) ).

% cInf_atLeastAtMost
tff(fact_4221_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Xa,Ya)) = Xa ) ) ) ).

% Inf_atLeastAtMost
tff(fact_4222_cInf__singleton,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ) ).

% cInf_singleton
tff(fact_4223_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Xa,Ya)) = Xa ) ) ) ).

% Inf_atLeastLessThan
tff(fact_4224_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Ya,Xa)) = Ya ) ) ) ).

% cInf_atLeastLessThan
tff(fact_4225_Inf__insert,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ).

% Inf_insert
tff(fact_4226_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atMost(A),Xa)) = bot_bot(A) ) ).

% Inf_atMost
tff(fact_4227_Inf__finite__insert,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ).

% Inf_finite_insert
tff(fact_4228_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),Z: A] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ) ).

% Inf_greatest
tff(fact_4229_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: A,A4: set(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(set(A),A,complete_Inf_Inf(A),A4))
        <=> ! [X: A] :
              ( member(A,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),X) ) ) ) ).

% le_Inf_iff
tff(fact_4230_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A4: set(A),V2: A] :
          ( member(A,U,A4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),V2) ) ) ) ).

% Inf_lower2
tff(fact_4231_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,A4: set(A)] :
          ( member(A,Xa,A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),Xa) ) ) ).

% Inf_lower
tff(fact_4232_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: set(A),A4: set(A)] :
          ( ! [B5: A] :
              ( member(A,B5,B2)
             => ? [X2: A] :
                  ( member(A,X2,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B5) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ).

% Inf_mono
tff(fact_4233_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),Xa: A] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),I2) )
         => ( ! [Y: A] :
                ( ! [I3: A] :
                    ( member(A,I3,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),I3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A4) = Xa ) ) ) ) ).

% Inf_eqI
tff(fact_4234_Inter__greatest,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(A)] :
      ( ! [X6: set(A)] :
          ( member(set(A),X6,A4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),X6) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)) ) ).

% Inter_greatest
tff(fact_4235_Inter__lower,axiom,
    ! [A: $tType,B2: set(A),A4: set(set(A))] :
      ( member(set(A),B2,A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)),B2) ) ).

% Inter_lower
tff(fact_4236_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X4: set(A)] :
          ( member(A,Z,X4)
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X3) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X4) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_4237_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X4: set(A),A3: A] :
          ( ! [X3: A] :
              ( member(A,X3,X4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X3) )
         => ( ! [Y: A] :
                ( ! [X2: A] :
                    ( member(A,X2,X4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X2) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A3) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X4) = A3 ) ) ) ) ).

% cInf_eq
tff(fact_4238_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S: set(A),A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S)),A3)
        <=> ? [X: A] :
              ( member(A,X,S)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3) ) ) ) ).

% Inf_less_iff
tff(fact_4239_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),Xa)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y4)
             => ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y4) ) ) ) ) ).

% Inf_le_iff
tff(fact_4240_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),U: A] :
          ( ! [V3: A] :
              ( member(A,V3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V3),U) )
         => ( ( A4 != bot_bot(set(A)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),U) ) ) ) ).

% Inf_less_eq
tff(fact_4241_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),A3: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X3) )
           => ( ! [Y: A] :
                  ( ! [X2: A] :
                      ( member(A,X2,X4)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X2) )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A3) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X4) = A3 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_4242_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),Z: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),X4)) ) ) ) ).

% cInf_greatest
tff(fact_4243_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( member(A,Xa,X4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X4)),Xa) ) ) ) ).

% cInf_le_finite
tff(fact_4244_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: set(A),A4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ).

% Inf_superset_mono
tff(fact_4245_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Z: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X4)),Z)
           => ? [X3: A] :
                ( member(A,X3,X4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Z) ) ) ) ) ).

% cInf_lessD
tff(fact_4246_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Xa: A,A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( member(A,Xa,X4)
           => ( ! [X3: A] :
                  ( member(A,X3,X4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X3) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X4)) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_4247_Inf__union__distrib,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: set(A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ).

% Inf_union_distrib
tff(fact_4248_Inter__anti__mono,axiom,
    ! [A: $tType,B2: set(set(A)),A4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),B2),A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B2)) ) ).

% Inter_anti_mono
tff(fact_4249_Inter__subset,axiom,
    ! [A: $tType,A4: set(set(A)),B2: set(A)] :
      ( ! [X6: set(A)] :
          ( member(set(A),X6,A4)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),B2) )
     => ( ( A4 != bot_bot(set(set(A))) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)),B2) ) ) ).

% Inter_subset
tff(fact_4250_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Inf_Inf(A),X4))
            <=> ! [X: A] :
                  ( member(A,X,X4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_4251_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ).

% Inf_le_Sup
tff(fact_4252_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S))),A3) ) ) ) ).

% cInf_abs_ge
tff(fact_4253_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: set(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ).

% less_eq_Inf_inter
tff(fact_4254_finite__Inf__in,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] :
                  ( member(A,X3,A4)
                 => ( member(A,Y,A4)
                   => member(A,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y),A4) ) )
             => member(A,aa(set(A),A,complete_Inf_Inf(A),A4),A4) ) ) ) ) ).

% finite_Inf_in
tff(fact_4255_cInf__eq__Inf__fin,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X4) = aa(set(A),A,lattic7752659483105999362nf_fin(A),X4) ) ) ) ) ).

% cInf_eq_Inf_fin
tff(fact_4256_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = aa(set(A),A,complete_Inf_Inf(A),A4) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_4257_Inter__Un__subset,axiom,
    ! [A: $tType,A4: set(set(A)),B2: set(set(A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A4)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B2))),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))),A4),B2))) ).

% Inter_Un_subset
tff(fact_4258_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S: set(A),L: A,E2: A] :
          ( ( S != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X3),L))),E2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(set(A),A,complete_Inf_Inf(A),S)),L))),E2) ) ) ) ).

% cInf_asclose
tff(fact_4259_mlex__eq,axiom,
    ! [A: $tType,F2: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F2,R) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_ja(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),F2),R))) ).

% mlex_eq
tff(fact_4260_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A4: set(A),R2: A,S3: A] :
          ( strict_mono_on(A,B,F2,A4)
         => ( member(A,R2,A4)
           => ( member(A,S3,A4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R2),S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R2)),aa(A,B,F2,S3)) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_4261_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [R3: A,S4: A] :
              ( member(A,R3,A4)
             => ( member(A,S4,A4)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),R3),S4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R3)),aa(A,B,F2,S4)) ) ) )
         => strict_mono_on(A,B,F2,A4) ) ) ).

% strict_mono_onI
tff(fact_4262_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( strict_mono_on(A,B,F2,A4)
        <=> ! [R5: A,S7: A] :
              ( ( member(A,R5,A4)
                & member(A,S7,A4)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),R5),S7) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,R5)),aa(A,B,F2,S7)) ) ) ) ).

% strict_mono_on_def
tff(fact_4263_Inf__nat__def1,axiom,
    ! [K4: set(nat)] :
      ( ( K4 != bot_bot(set(nat)) )
     => member(nat,aa(set(nat),nat,complete_Inf_Inf(nat),K4),K4) ) ).

% Inf_nat_def1
tff(fact_4264_mlex__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xa: A,Ya: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Ya))
     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),mlex_prod(A,F2,R)) ) ).

% mlex_less
tff(fact_4265_mlex__iff,axiom,
    ! [A: $tType,Xa: A,Ya: A,F2: fun(A,nat),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),mlex_prod(A,F2,R))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Ya))
        | ( ( aa(A,nat,F2,Xa) = aa(A,nat,F2,Ya) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R) ) ) ) ).

% mlex_iff
tff(fact_4266_mlex__leq,axiom,
    ! [A: $tType,F2: fun(A,nat),Xa: A,Ya: A,R: set(product_prod(A,A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Ya))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R)
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),mlex_prod(A,F2,R)) ) ) ).

% mlex_leq
tff(fact_4267_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F2: fun(A,B),A4: set(A),Xa: A,Ya: A] :
          ( strict_mono_on(A,B,F2,A4)
         => ( member(A,Xa,A4)
           => ( member(A,Ya,A4)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya)) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_4268_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jb(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_apsnd(code_integer,code_integer,code_integer,uminus_uminus(code_integer)),
              $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_jc(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_4269_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,Xa,Xs)) = aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_4270_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,Na,Xs)),M) = aa(A,product_prod(nat,A),product_Pair(nat,A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_4271_semiring__char__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: itself(A)] : semiri4206861660011772517g_char(A,Uu) = gcd_Gcd(nat,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_jd(nat,$o))) ) ).

% semiring_char_def
tff(fact_4272_set__remove1__subset,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,Xa,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_remove1_subset
tff(fact_4273_sub__num__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,bit0(L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_4274_div__add__self2__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xa: A,B3: B,A3: B] :
          ( nO_MATCH(A,B,Xa,B3)
         => ( ( B3 != zero_zero(B) )
           => ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B3),B3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A3,B3)),one_one(B)) ) ) ) ) ).

% div_add_self2_no_field
tff(fact_4275_div__add__self1__no__field,axiom,
    ! [A: $tType,B: $tType] :
      ( ( euclid4440199948858584721cancel(B)
        & field(A) )
     => ! [Xa: A,B3: B,A3: B] :
          ( nO_MATCH(A,B,Xa,B3)
         => ( ( B3 != zero_zero(B) )
           => ( divide_divide(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),B3),A3),B3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),divide_divide(B,A3,B3)),one_one(B)) ) ) ) ) ).

% div_add_self1_no_field
tff(fact_4276_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_je(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F2),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_4277_Suc__funpow,axiom,
    ! [Na: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),Na),suc) = aa(nat,fun(nat,nat),plus_plus(nat),Na) ).

% Suc_funpow
tff(fact_4278_funpow__0,axiom,
    ! [A: $tType,F2: fun(A,A),Xa: 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),Xa) = Xa ).

% funpow_0
tff(fact_4279_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_4280_sub__num__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,bit0(K),one2) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% sub_num_simps(4)
tff(fact_4281_funpow__swap1,axiom,
    ! [A: $tType,F2: fun(A,A),Na: nat,Xa: 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)),Na),F2),Xa)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F2),aa(A,A,F2,Xa)) ).

% funpow_swap1
tff(fact_4282_funpow__mult,axiom,
    ! [A: $tType,Na: 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)),Na),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),Na)),F2) ).

% funpow_mult
tff(fact_4283_eval__nat__numeral_I2_J,axiom,
    ! [Na: num] : aa(num,nat,numeral_numeral(nat),bit0(Na)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(Na))) ).

% eval_nat_numeral(2)
tff(fact_4284_one__plus__BitM,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(Na)) = bit0(Na) ).

% one_plus_BitM
tff(fact_4285_BitM__plus__one,axiom,
    ! [Na: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(Na)),one2) = bit0(Na) ).

% BitM_plus_one
tff(fact_4286_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : aa(nat,A,semiring_1_of_nat(A),Na) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_4287_numeral__add__unfold__funpow,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),K)),A3) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(num,nat,numeral_numeral(nat),K)),aa(A,fun(A,A),plus_plus(A),one_one(A))),A3) ) ).

% numeral_add_unfold_funpow
tff(fact_4288_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_4289_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(num,A,numeral_numeral(A),bitM(Na)) = aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),bit0(Na))),one_one(A)) ) ).

% numeral_BitM
tff(fact_4290_relpowp__bot,axiom,
    ! [A: $tType,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),bot_bot(fun(A,fun(A,$o)))) = bot_bot(fun(A,fun(A,$o))) ) ) ).

% relpowp_bot
tff(fact_4291_relpowp__fun__conv,axiom,
    ! [A: $tType,Na: nat,P: fun(A,fun(A,$o)),Xa: A,Ya: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P),Xa),Ya)
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = Xa )
          & ( aa(nat,A,F7,Na) = Ya )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
             => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,F7,I4)),aa(nat,A,F7,aa(nat,nat,suc,I4))) ) ) ) ).

% relpowp_fun_conv
tff(fact_4292_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_4293_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),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_4294_relpowp_Osimps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),R) = fequal(A) ).

% relpowp.simps(1)
tff(fact_4295_relpowp__0__E,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xa: A,Ya: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P),Xa),Ya)
     => ( Xa = Ya ) ) ).

% relpowp_0_E
tff(fact_4296_relpowp__0__I,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xa: A] : aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),zero_zero(nat)),P),Xa),Xa) ).

% relpowp_0_I
tff(fact_4297_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_4298_relpowp__E,axiom,
    ! [A: $tType,Na: nat,P: fun(A,fun(A,$o)),Xa: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P),Xa),Z)
     => ( ( ( Na = zero_zero(nat) )
         => ( Xa != Z ) )
       => ~ ! [Y: A,M4: nat] :
              ( ( Na = aa(nat,nat,suc,M4) )
             => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M4),P),Xa),Y)
               => ~ aa(A,$o,aa(A,fun(A,$o),P,Y),Z) ) ) ) ) ).

% relpowp_E
tff(fact_4299_relpowp__E2,axiom,
    ! [A: $tType,Na: nat,P: fun(A,fun(A,$o)),Xa: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),Na),P),Xa),Z)
     => ( ( ( Na = zero_zero(nat) )
         => ( Xa != Z ) )
       => ~ ! [Y: A,M4: nat] :
              ( ( Na = aa(nat,nat,suc,M4) )
             => ( aa(A,$o,aa(A,fun(A,$o),P,Xa),Y)
               => ~ aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(nat,fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),compow(fun(A,fun(A,$o))),M4),P),Y),Z) ) ) ) ) ).

% relpowp_E2
tff(fact_4300_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,aa(fun(code_integer,fun(code_integer,nat)),fun(product_prod(code_integer,code_integer),nat),product_case_prod(code_integer,code_integer,nat),aTP_Lamp_jf(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% nat_of_integer_code
tff(fact_4301_Fpow__Pow__finite,axiom,
    ! [A: $tType,A4: set(A)] : finite_Fpow(A,A4) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),pow2(A,A4)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A))) ).

% Fpow_Pow_finite
tff(fact_4302_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_jg(A,fun(A,$o)),aTP_Lamp_jh(A,fun(A,$o))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_4303_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_ag(nat,fun(nat,$o)),aTP_Lamp_af(nat,fun(nat,$o))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_4304_semilattice__neutr__order_Oeq__neutr__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => ( ( aa(A,A,aa(A,fun(A,A),F2,A3),B3) = Z )
      <=> ( ( A3 = Z )
          & ( B3 = Z ) ) ) ) ).

% semilattice_neutr_order.eq_neutr_iff
tff(fact_4305_semilattice__neutr__order_Oneutr__eq__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Z: A,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( semila1105856199041335345_order(A,F2,Z,Less_eq,Less)
     => ( ( Z = aa(A,A,aa(A,fun(A,A),F2,A3),B3) )
      <=> ( ( A3 = Z )
          & ( B3 = Z ) ) ) ) ).

% semilattice_neutr_order.neutr_eq_iff
tff(fact_4306_empty__in__Fpow,axiom,
    ! [A: $tType,A4: set(A)] : member(set(A),bot_bot(set(A)),finite_Fpow(A,A4)) ).

% empty_in_Fpow
tff(fact_4307_Fpow__not__empty,axiom,
    ! [A: $tType,A4: set(A)] : finite_Fpow(A,A4) != bot_bot(set(set(A))) ).

% Fpow_not_empty
tff(fact_4308_Fpow__mono,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A4)),finite_Fpow(A,B2)) ) ).

% Fpow_mono
tff(fact_4309_Fpow__subset__Pow,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),finite_Fpow(A,A4)),pow2(A,A4)) ).

% Fpow_subset_Pow
tff(fact_4310_Fpow__def,axiom,
    ! [A: $tType,A4: set(A)] : finite_Fpow(A,A4) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_ji(set(A),fun(set(A),$o),A4)) ).

% Fpow_def
tff(fact_4311_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,aa(fun(code_integer,fun(code_integer,int)),fun(product_prod(code_integer,code_integer),int),product_case_prod(code_integer,code_integer,int),aTP_Lamp_jj(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ) ).

% int_of_integer_code
tff(fact_4312_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Na: nat] :
      ( ! [X3: list(A)] :
          ( member(list(A),X3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X3) = Na ) )
     => ( 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)),Na) ) ) ).

% length_mul_elem
tff(fact_4313_eq__numeral__iff__iszero_I8_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ya: num] :
          ( ( one_one(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ya)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Ya))) ) ) ).

% eq_numeral_iff_iszero(8)
tff(fact_4314_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_4315_iszero__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W2: num] :
          ( ring_1_iszero(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W2)))
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),W2)) ) ) ).

% iszero_neg_numeral
tff(fact_4316_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_4317_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_4318_not__iszero__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [W2: num] : ~ ring_1_iszero(A,aa(num,A,numeral_numeral(A),W2)) ) ).

% not_iszero_numeral
tff(fact_4319_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_4320_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa = Ya )
        <=> ring_1_iszero(A,aa(A,A,minus_minus(A,Xa),Ya)) ) ) ).

% eq_iff_iszero_diff
tff(fact_4321_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_4322_less__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),Xa),Xaa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),code_int_of_integer(Xa)),code_int_of_integer(Xaa)) ) ).

% less_integer.rep_eq
tff(fact_4323_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_4324_less__eq__integer_Orep__eq,axiom,
    ! [Xa: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),Xa),Xaa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),code_int_of_integer(Xa)),code_int_of_integer(Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_4325_eq__numeral__iff__iszero_I10_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ya: num] :
          ( ( zero_zero(A) = aa(num,A,numeral_numeral(A),Ya) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Ya)) ) ) ).

% eq_numeral_iff_iszero(10)
tff(fact_4326_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num] :
          ( ( aa(num,A,numeral_numeral(A),Xa) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xa)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_4327_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_4328_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_4329_eq__numeral__iff__iszero_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num,Ya: num] :
          ( ( aa(num,A,numeral_numeral(A),Xa) = aa(num,A,numeral_numeral(A),Ya) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Xa,Ya)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_4330_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Xa)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_4331_eq__numeral__iff__iszero_I12_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ya: num] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ya)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),Ya)) ) ) ).

% eq_numeral_iff_iszero(12)
tff(fact_4332_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_4333_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num,Ya: num] :
          ( ( aa(num,A,numeral_numeral(A),Xa) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ya)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),Ya))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_4334_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num,Ya: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa)) = aa(num,A,numeral_numeral(A),Ya) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),Ya))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_4335_eq__numeral__iff__iszero_I4_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num,Ya: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xa)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Ya)) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Ya,Xa)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_4336_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys3: list(A)] :
            ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs2: list(A)] :
              ( member(list(A),Ys3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
             => ( member(list(A),Zs2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
               => ( ( Ys3 != Zs2 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys3)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_4337_eq__numeral__iff__iszero_I6_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Ya: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Ya) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,one2,Ya)) ) ) ).

% eq_numeral_iff_iszero(6)
tff(fact_4338_eq__numeral__iff__iszero_I5_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xa: num] :
          ( ( aa(num,A,numeral_numeral(A),Xa) = one_one(A) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Xa,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_4339_set__n__lists,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Na,Xs)) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(list(A),fun(list(A),$o),aTP_Lamp_jk(nat,fun(list(A),fun(list(A),$o)),Na),Xs)) ).

% set_n_lists
tff(fact_4340_prod_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P3: fun(A,B),I: A] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bu(set(A),fun(fun(A,B),fun(A,$o)),I5),P3)))
         => ( groups1962203154675924110t_prod(A,B,P3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),I),I5)) = $ite(member(A,I,I5),groups1962203154675924110t_prod(A,B,P3,I5),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,P3,I)),groups1962203154675924110t_prod(A,B,P3,I5))) ) ) ) ).

% prod.insert'
tff(fact_4341_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))) = remove1(A,Xa,linord4507533701916653071of_set(A,A4)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_4342_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),removeAll(list(A),nil(A),Xs))
        & ! [Ys4: list(A)] :
            ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
           => distinct(A,Ys4) )
        & ! [Ys4: list(A),Zs3: list(A)] :
            ( ( member(list(A),Ys4,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & member(list(A),Zs3,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
              & ( Ys4 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys4)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_4343_set__empty2,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty2
tff(fact_4344_set__empty,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xs) = bot_bot(set(A)) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty
tff(fact_4345_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_4346_empty__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( nil(A) = replicate(A,Na,Xa) )
    <=> ( Na = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_4347_replicate__empty,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( replicate(A,Na,Xa) = nil(A) )
    <=> ( Na = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_4348_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A,bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_4349_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( linord4507533701916653071of_set(A,A4) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_4350_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(list(A),set(A),set2(A),linord4507533701916653071of_set(A,A4)) = A4 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_4351_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A)] : groups1962203154675924110t_prod(B,A,P3,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_4352_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_4353_prod_Oeq__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),P3: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1962203154675924110t_prod(A,B,P3,I5) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),P3),I5) ) ) ) ).

% prod.eq_sum
tff(fact_4354_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_4355_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( linord4507533701916653071of_set(A,A4) = nil(A) )
          <=> ( A4 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_4356_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B2: set(A)] :
          ( ( linord4507533701916653071of_set(A,A4) = linord4507533701916653071of_set(A,B2) )
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( A4 = B2 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_4357_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_4358_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_4359_replicate__0,axiom,
    ! [A: $tType,Xa: A] : replicate(A,zero_zero(nat),Xa) = nil(A) ).

% replicate_0
tff(fact_4360_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xa: fun(A,nat)] : size_list(A,Xa,nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_4361_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Ya: A] : aa(A,nat,count_list(A,nil(A)),Ya) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_4362_prod_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_jl(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ).

% prod.distrib_triv'
tff(fact_4363_prod_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X3) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,S) = groups1962203154675924110t_prod(A,B,G,T3) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_4364_prod_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X3) = one_one(B) ) )
           => ( groups1962203154675924110t_prod(A,B,G,T3) = groups1962203154675924110t_prod(A,B,G,S) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_4365_prod_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,H,I2) = one_one(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
             => ( groups1962203154675924110t_prod(A,B,G,S) = groups1962203154675924110t_prod(A,B,H,T3) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_4366_prod_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),T3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( ! [X3: A] :
                ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),T3),S))
               => ( aa(A,B,G,X3) = one_one(B) ) )
           => ( ! [X3: A] :
                  ( member(A,X3,S)
                 => ( aa(A,B,G,X3) = aa(A,B,H,X3) ) )
             => ( groups1962203154675924110t_prod(A,B,G,T3) = groups1962203154675924110t_prod(A,B,H,S) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_4367_prod_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bu(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_bu(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1962203154675924110t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_jl(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),times_times(B),groups1962203154675924110t_prod(A,B,G,I5)),groups1962203154675924110t_prod(A,B,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_4368_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P3: fun(B,A),I5: set(B)] :
          groups1962203154675924110t_prod(B,A,P3,I5) = $ite(aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_jm(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P3),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aTP_Lamp_jm(fun(B,A),fun(set(B),fun(B,$o)),P3),I5))),one_one(A)) ) ).

% prod.G_def
tff(fact_4369_Pow__set_I1_J,axiom,
    ! [A: $tType] : pow2(A,aa(list(A),set(A),set2(A),nil(A))) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_4370_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = linorder_insort_key(A,A,aTP_Lamp_jn(A,A),Xa,linord4507533701916653071of_set(A,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_4371_rat__floor__lemma,axiom,
    ! [A3: int,B3: int] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),divide_divide(int,A3,B3))),fract(A3,B3))
      & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A3,B3)),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A3,B3)),one_one(int)))) ) ).

% rat_floor_lemma
tff(fact_4372_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C3: nat,Xa: nat,Ya: nat] :
      aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_jo(nat,fun(nat,nat),C3)),set_or7035219750837199246ssThan(nat,Xa,Ya)) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),C3),Ya),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,Xa),C3),aa(nat,nat,minus_minus(nat,Ya),C3)),
        $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ya),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_4373_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,aa(fun(code_integer,fun(code_integer,num)),fun(product_prod(code_integer,code_integer),num),product_case_prod(code_integer,code_integer,num),aTP_Lamp_jp(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% num_of_integer_code
tff(fact_4374_image__eqI,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),Xa: B,A4: set(B)] :
      ( ( B3 = aa(B,A,F2,Xa) )
     => ( member(B,Xa,A4)
       => member(A,B3,aa(set(B),set(A),image2(B,A,F2),A4)) ) ) ).

% image_eqI
tff(fact_4375_image__ident,axiom,
    ! [A: $tType,Y3: set(A)] : aa(set(A),set(A),image2(A,A,aTP_Lamp_jq(A,A)),Y3) = Y3 ).

% image_ident
tff(fact_4376_image__is__empty,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),A4) = bot_bot(set(A)) )
    <=> ( A4 = bot_bot(set(B)) ) ) ).

% image_is_empty
tff(fact_4377_empty__is__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(B),set(A),image2(B,A,F2),A4) )
    <=> ( A4 = bot_bot(set(B)) ) ) ).

% empty_is_image
tff(fact_4378_image__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B))) = bot_bot(set(A)) ).

% image_empty
tff(fact_4379_finite__imageI,axiom,
    ! [B: $tType,A: $tType,F3: set(A),H: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,H),F3)) ) ).

% finite_imageI
tff(fact_4380_insert__image,axiom,
    ! [B: $tType,A: $tType,Xa: A,A4: set(A),F2: fun(A,B)] :
      ( member(A,Xa,A4)
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,Xa)),aa(set(A),set(B),image2(A,B,F2),A4)) = aa(set(A),set(B),image2(A,B,F2),A4) ) ) ).

% insert_image
tff(fact_4381_image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A3: B,B2: set(B)] : aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),B2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,F2,A3)),aa(set(B),set(A),image2(B,A,F2),B2)) ).

% image_insert
tff(fact_4382_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S: set(A)] : aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S) = S ) ).

% image_add_0
tff(fact_4383_ccSUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_jr(B,A)),A4)) = bot_bot(A) ) ).

% ccSUP_bot
tff(fact_4384_SUP__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_js(B,A)),A4)) = bot_bot(A) ) ).

% SUP_bot
tff(fact_4385_SUP__bot__conv_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B2),A4)) = bot_bot(A) )
        <=> ! [X: B] :
              ( member(B,X,A4)
             => ( aa(B,A,B2,X) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(1)
tff(fact_4386_SUP__bot__conv_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: fun(B,A),A4: set(B)] :
          ( ( bot_bot(A) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,B2),A4)) )
        <=> ! [X: B] :
              ( member(B,X,A4)
             => ( aa(B,A,B2,X) = bot_bot(A) ) ) ) ) ).

% SUP_bot_conv(2)
tff(fact_4387_ccSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),F2: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_jt(B,fun(A,B),F2)),A4)) = F2 ) ) ) ).

% ccSUP_const
tff(fact_4388_SUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ju(B,fun(A,B),F2)),A4)) = F2 ) ) ) ).

% SUP_const
tff(fact_4389_cSUP__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),C3: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_jv(B,fun(A,B),C3)),A4)) = C3 ) ) ) ).

% cSUP_const
tff(fact_4390_ccINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),F2: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_jt(B,fun(A,B),F2)),A4)) = F2 ) ) ) ).

% ccINF_const
tff(fact_4391_cINF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),C3: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_jv(B,fun(A,B),C3)),A4)) = C3 ) ) ) ).

% cINF_const
tff(fact_4392_INF__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_ju(B,fun(A,B),F2)),A4)) = F2 ) ) ) ).

% INF_const
tff(fact_4393_if__image__distrib,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F2: fun(B,A),G: fun(B,A),S: set(B)] : aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jw(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),F2),G)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),P)))),aa(set(B),set(A),image2(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S),aa(fun(B,$o),set(B),collect(B),aTP_Lamp_jx(fun(B,$o),fun(B,$o),P))))) ).

% if_image_distrib
tff(fact_4394_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4)) = bot_bot(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),bot_bot(A)),X)
             => ? [Xa3: B] :
                  ( member(B,Xa3,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,Xa3)),X) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_4395_ccSUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).

% ccSUP_empty
tff(fact_4396_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( linord4507533701916653071of_set(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = linorder_insort_key(A,A,aTP_Lamp_jn(A,A),Xa,linord4507533701916653071of_set(A,A4)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_4397_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B3)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_4398_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),D2)
         => ( aa(set(A),set(A),image2(A,A,aTP_Lamp_jy(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,divide_divide(A,A3,D2),divide_divide(A,B3,D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_4399_less__rat,axiom,
    ! [B3: int,D2: int,A3: int,C3: int] :
      ( ( B3 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A3,B3)),fract(C3,D2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D2))) ) ) ) ).

% less_rat
tff(fact_4400_le__rat,axiom,
    ! [B3: int,D2: int,A3: int,C3: int] :
      ( ( B3 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),fract(A3,B3)),fract(C3,D2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),C3),B3)),aa(int,int,aa(int,fun(int,int),times_times(int),B3),D2))) ) ) ) ).

% le_rat
tff(fact_4401_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A4)),B2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow2(B,A4))),pow2(A,B2)) ) ).

% image_Pow_mono
tff(fact_4402_image__Pow__surj,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B),B2: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),A4) = B2 )
     => ( aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),pow2(B,A4)) = pow2(A,B2) ) ) ).

% image_Pow_surj
tff(fact_4403_image__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),B2: set(B)] : aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(B),set(A),image2(B,A,F2),A4)),aa(set(B),set(A),image2(B,A,F2),B2)) ).

% image_Un
tff(fact_4404_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),A4))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),A4)
          & ( B2 = aa(set(B),set(A),image2(B,A,F2),AA) ) ) ) ).

% subset_image_iff
tff(fact_4405_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A4)),B2)
    <=> ! [X: B] :
          ( member(B,X,A4)
         => member(A,aa(B,A,F2,X),B2) ) ) ).

% image_subset_iff
tff(fact_4406_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),A4))
     => ~ ! [C7: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C7),A4)
           => ( B2 != aa(set(B),set(A),image2(B,A,F2),C7) ) ) ) ).

% subset_imageE
tff(fact_4407_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(A,B),B2: set(B)] :
      ( ! [X3: A] :
          ( member(A,X3,A4)
         => member(B,aa(A,B,F2,X3),B2) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),B2) ) ).

% image_subsetI
tff(fact_4408_image__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),aa(set(A),set(B),image2(A,B,F2),B2)) ) ).

% image_mono
tff(fact_4409_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),P: fun(set(A),$o)] :
      ( ! [B11: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F2),A4))
         => aa(set(A),$o,P,B11) )
    <=> ! [B11: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A4)
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),B11)) ) ) ).

% all_subset_image
tff(fact_4410_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(A,B),B2: set(B)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
         => member(B,aa(A,B,F2,X3),B2) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(fun(A,$o),set(A),collect(A),P))),B2) ) ).

% image_Collect_subsetI
tff(fact_4411_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,F2),A4))
       => ? [X3: A] :
            ( member(A,X3,A4)
            & ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_jz(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A4),F2),X3))) ) ) ) ).

% pigeonhole_infinite
tff(fact_4412_imageE,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),A4: set(B)] :
      ( member(A,B3,aa(set(B),set(A),image2(B,A,F2),A4))
     => ~ ! [X3: B] :
            ( ( B3 = aa(B,A,F2,X3) )
           => ~ member(B,X3,A4) ) ) ).

% imageE
tff(fact_4413_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),A4: set(C)] : aa(set(B),set(A),image2(B,A,F2),aa(set(C),set(B),image2(C,B,G),A4)) = aa(set(C),set(A),image2(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_ka(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),A4) ).

% image_image
tff(fact_4414_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B),P: fun(A,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_kb(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F2),A4),P)) = aa(set(B),set(A),image2(B,A,F2),aa(fun(B,$o),set(B),collect(B),aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_kc(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F2),A4),P))) ).

% Compr_image_eq
tff(fact_4415_rev__image__eqI,axiom,
    ! [B: $tType,A: $tType,Xa: A,A4: set(A),B3: B,F2: fun(A,B)] :
      ( member(A,Xa,A4)
     => ( ( B3 = aa(A,B,F2,Xa) )
       => member(B,B3,aa(set(A),set(B),image2(A,B,F2),A4)) ) ) ).

% rev_image_eqI
tff(fact_4416_ball__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),P: fun(A,$o)] :
      ( ! [X3: A] :
          ( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A4))
         => aa(A,$o,P,X3) )
     => ! [X2: B] :
          ( member(B,X2,A4)
         => aa(A,$o,P,aa(B,A,F2,X2)) ) ) ).

% ball_imageD
tff(fact_4417_image__cong,axiom,
    ! [B: $tType,A: $tType,M5: set(A),N4: set(A),F2: fun(A,B),G: fun(A,B)] :
      ( ( M5 = N4 )
     => ( ! [X3: A] :
            ( member(A,X3,N4)
           => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) )
       => ( aa(set(A),set(B),image2(A,B,F2),M5) = aa(set(A),set(B),image2(A,B,G),N4) ) ) ) ).

% image_cong
tff(fact_4418_bex__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),P: fun(A,$o)] :
      ( ? [X2: A] :
          ( member(A,X2,aa(set(B),set(A),image2(B,A,F2),A4))
          & aa(A,$o,P,X2) )
     => ? [X3: B] :
          ( member(B,X3,A4)
          & aa(A,$o,P,aa(B,A,F2,X3)) ) ) ).

% bex_imageD
tff(fact_4419_image__iff,axiom,
    ! [A: $tType,B: $tType,Z: A,F2: fun(B,A),A4: set(B)] :
      ( member(A,Z,aa(set(B),set(A),image2(B,A,F2),A4))
    <=> ? [X: B] :
          ( member(B,X,A4)
          & ( Z = aa(B,A,F2,X) ) ) ) ).

% image_iff
tff(fact_4420_imageI,axiom,
    ! [B: $tType,A: $tType,Xa: A,A4: set(A),F2: fun(A,B)] :
      ( member(A,Xa,A4)
     => member(B,aa(A,B,F2,Xa),aa(set(A),set(B),image2(A,B,F2),A4)) ) ).

% imageI
tff(fact_4421_UNION__singleton__eq__range,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_kd(fun(B,A),fun(B,set(A)),F2)),A4)) = aa(set(B),set(A),image2(B,A,F2),A4) ).

% UNION_singleton_eq_range
tff(fact_4422_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A4)),B2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image2(set(B),set(A),image2(B,A,F2)),finite_Fpow(B,A4))),finite_Fpow(A,B2)) ) ).

% image_Fpow_mono
tff(fact_4423_SUP__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(A),B2: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ? [X2: B] :
                  ( member(B,X2,B2)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,I2)),aa(B,C,G,X2)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B2)
               => ? [X2: A] :
                    ( member(A,X2,A4)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,J2)),aa(A,C,F2,X2)) ) )
           => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,F2),A4)) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B2)) ) ) ) ) ).

% SUP_eq
tff(fact_4424_INF__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(A),B2: set(B),G: fun(B,C),F2: fun(A,C)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ? [X2: B] :
                  ( member(B,X2,B2)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,G,X2)),aa(A,C,F2,I2)) ) )
         => ( ! [J2: B] :
                ( member(B,J2,B2)
               => ? [X2: A] :
                    ( member(A,X2,A4)
                    & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X2)),aa(B,C,G,J2)) ) )
           => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,F2),A4)) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),B2)) ) ) ) ) ).

% INF_eq
tff(fact_4425_zero__notin__Suc__image,axiom,
    ! [A4: set(nat)] : ~ member(nat,zero_zero(nat),aa(set(nat),set(nat),image2(nat,nat,suc),A4)) ).

% zero_notin_Suc_image
tff(fact_4426_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),P: fun(set(A),$o)] :
      ( ! [B11: set(A)] :
          ( ( aa(set(A),$o,finite_finite2(A),B11)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F2),A4)) )
         => aa(set(A),$o,P,B11) )
    <=> ! [B11: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),B11)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A4) )
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),B11)) ) ) ).

% all_finite_subset_image
tff(fact_4427_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),P: fun(set(A),$o)] :
      ( ? [B11: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),B11)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image2(B,A,F2),A4))
          & aa(set(A),$o,P,B11) )
    <=> ? [B11: set(B)] :
          ( aa(set(B),$o,finite_finite2(B),B11)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A4)
          & aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),B11)) ) ) ).

% ex_finite_subset_image
tff(fact_4428_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),A4))
       => ? [C7: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C7),A4)
            & aa(set(B),$o,finite_finite2(B),C7)
            & ( B2 = aa(set(B),set(A),image2(B,A,F2),C7) ) ) ) ) ).

% finite_subset_image
tff(fact_4429_finite__surj,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),image2(A,B,F2),A4))
       => aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% finite_surj
tff(fact_4430_SUP__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => ( aa(A,B,F2,I2) = Xa ) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),I5)) = Xa ) ) ) ) ).

% SUP_eq_const
tff(fact_4431_INF__eq__const,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => ( aa(A,B,F2,I2) = Xa ) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I5)) = Xa ) ) ) ) ).

% INF_eq_const
tff(fact_4432_finite__image__absD,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),image2(A,A,abs_abs(A)),S))
         => aa(set(A),$o,finite_finite2(A),S) ) ) ).

% finite_image_absD
tff(fact_4433_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),B2: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image2(B,A,F2),A4)),aa(set(B),set(A),image2(B,A,F2),B2))) ).

% image_Int_subset
tff(fact_4434_sup__Inf,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,B2: set(A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B2)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),sup_sup(A),A3)),B2)) ) ).

% sup_Inf
tff(fact_4435_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B),B2: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),aa(set(B),set(A),image2(B,A,F2),A4)),aa(set(B),set(A),image2(B,A,F2),B2))),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),minus_minus(set(B),A4),B2))) ).

% image_diff_subset
tff(fact_4436_Rat__induct__pos,axiom,
    ! [P: fun(rat,$o),Q5: rat] :
      ( ! [A5: int,B5: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
         => aa(rat,$o,P,fract(A5,B5)) )
     => aa(rat,$o,P,Q5) ) ).

% Rat_induct_pos
tff(fact_4437_set__insort__key,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Xs: list(A)] : aa(list(A),set(A),set2(A),linorder_insort_key(A,B,F2,Xa,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_key
tff(fact_4438_SUP__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: fun(A,B),Xa: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),Xa) )
         => ( ! [Y: B] :
                ( ! [I3: A] :
                    ( member(A,I3,A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I3)),Y) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),Y) )
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4)) = Xa ) ) ) ) ).

% SUP_eqI
tff(fact_4439_SUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [A4: set(A),B2: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( ! [N2: A] :
              ( member(A,N2,A4)
             => ? [X2: B] :
                  ( member(B,X2,B2)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,N2)),aa(B,C,G,X2)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,F2),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B2))) ) ) ).

% SUP_mono
tff(fact_4440_SUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: fun(A,B),U: B] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),U) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),U) ) ) ).

% SUP_least
tff(fact_4441_SUP__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A4: set(A)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),A4))) ) ) ).

% SUP_mono'
tff(fact_4442_SUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F2: fun(A,B)] :
          ( member(A,I,A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ).

% SUP_upper
tff(fact_4443_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A4: set(B),U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4))),U)
        <=> ! [X: B] :
              ( member(B,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X)),U) ) ) ) ).

% SUP_le_iff
tff(fact_4444_SUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),U: B,F2: fun(A,B)] :
          ( member(A,I,A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ).

% SUP_upper2
tff(fact_4445_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: A,F2: fun(B,A),A4: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4)))
        <=> ? [X: B] :
              ( member(B,X,A4)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(B,A,F2,X)) ) ) ) ).

% less_SUP_iff
tff(fact_4446_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A4: set(B),Ya: A,I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4))),Ya)
         => ( member(B,I,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I)),Ya) ) ) ) ).

% SUP_lessD
tff(fact_4447_INF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),U: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ).

% INF_greatest
tff(fact_4448_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F2: fun(B,A),A4: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4)))
        <=> ! [X: B] :
              ( member(B,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F2,X)) ) ) ) ).

% le_INF_iff
tff(fact_4449_INF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F2: fun(A,B),U: B] :
          ( member(A,I,A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),U)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),U) ) ) ) ).

% INF_lower2
tff(fact_4450_INF__mono_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [F2: fun(A,B),G: fun(A,B),A4: set(A)] :
          ( ! [X3: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3))
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),A4))) ) ) ).

% INF_mono'
tff(fact_4451_INF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I: A,A4: set(A),F2: fun(A,B)] :
          ( member(A,I,A4)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(A,B,F2,I)) ) ) ).

% INF_lower
tff(fact_4452_INF__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comple6319245703460814977attice(C)
     => ! [B2: set(A),A4: set(B),F2: fun(B,C),G: fun(A,C)] :
          ( ! [M4: A] :
              ( member(A,M4,B2)
             => ? [X2: B] :
                  ( member(B,X2,A4)
                  & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(B,C,F2,X2)),aa(A,C,G,M4)) ) )
         => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F2),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,G),B2))) ) ) ).

% INF_mono
tff(fact_4453_INF__eqI,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),Xa: B,F2: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),aa(A,B,F2,I2)) )
         => ( ! [Y: B] :
                ( ! [I3: A] :
                    ( member(A,I3,A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),aa(A,B,F2,I3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Y),Xa) )
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4)) = Xa ) ) ) ) ).

% INF_eqI
tff(fact_4454_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B),A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4))),A3)
        <=> ? [X: B] :
              ( member(B,X,A4)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),A3) ) ) ) ).

% INF_less_iff
tff(fact_4455_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Ya: A,F2: fun(B,A),A4: set(B),I: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4)))
         => ( member(B,I,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),aa(B,A,F2,I)) ) ) ) ).

% less_INF_D
tff(fact_4456_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
    <=> ? [N: nat,F7: fun(nat,A)] : A4 = aa(set(nat),set(A),image2(nat,A,F7),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),N))) ) ).

% finite_conv_nat_seg_image
tff(fact_4457_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A4: set(A),F2: fun(nat,A),Na: nat] :
      ( ( A4 = aa(set(nat),set(A),image2(nat,A,F2),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Na))) )
     => aa(set(A),$o,finite_finite2(A),A4) ) ).

% nat_seg_image_imp_finite
tff(fact_4458_image__constant,axiom,
    ! [A: $tType,B: $tType,Xa: A,A4: set(A),C3: B] :
      ( member(A,Xa,A4)
     => ( aa(set(A),set(B),image2(A,B,aTP_Lamp_ke(B,fun(A,B),C3)),A4) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),C3),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_4459_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C3: A,A4: set(B)] :
      aa(set(B),set(A),image2(B,A,aTP_Lamp_kf(A,fun(B,A),C3)),A4) = $ite(A4 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),C3),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_4460_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S: set(A),H: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),H),S) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_kh(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H),G)),aa(set(A),set(C),image2(A,C,G),S)) ) ) ) ).

% sum.image_gen
tff(fact_4461_SUP__absorb,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [K: A,I5: set(A),A4: fun(A,B)] :
          ( member(A,K,I5)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,A4,K)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,A4),I5))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,A4),I5)) ) ) ) ).

% SUP_absorb
tff(fact_4462_complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A4: set(B),G: fun(B,A)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,G),A4))) = aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ki(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A4)) ) ).

% complete_lattice_class.SUP_sup_distrib
tff(fact_4463_INF__sup__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),A4: set(B),G: fun(C,A),B2: set(C)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,G),B2))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_kk(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F2),G),B2)),A4)) ) ).

% INF_sup_distrib2
tff(fact_4464_sup__INF,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: A,F2: fun(B,A),B2: set(B)] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),B2))) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kl(A,fun(fun(B,A),fun(B,A)),A3),F2)),B2)) ) ).

% sup_INF
tff(fact_4465_Inf__sup,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B2: set(A),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B2)),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_km(A,fun(A,A),A3)),B2)) ) ).

% Inf_sup
tff(fact_4466_INF__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [F2: fun(B,A),B2: set(B),A3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),B2))),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aa(A,fun(B,A),aTP_Lamp_kn(fun(B,A),fun(A,fun(B,A)),F2),A3)),B2)) ) ).

% INF_sup
tff(fact_4467_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S: set(A),H: fun(A,B),G: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),H),S) = aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7121269368397514597t_prod(C,B),aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ko(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S),H),G)),aa(set(A),set(C),image2(A,C,G),S)) ) ) ) ).

% prod.image_gen
tff(fact_4468_the__elem__image__unique,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B),Xa: A] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ! [Y: A] :
            ( member(A,Y,A4)
           => ( aa(A,B,F2,Y) = aa(A,B,F2,Xa) ) )
       => ( the_elem(B,aa(set(A),set(B),image2(A,B,F2),A4)) = aa(A,B,F2,Xa) ) ) ) ).

% the_elem_image_unique
tff(fact_4469_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [Xa: A,F2: fun(B,A),A4: set(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4)))
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xa)
             => ? [X: B] :
                  ( member(B,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),aa(B,A,F2,X)) ) ) ) ) ).

% le_SUP_iff
tff(fact_4470_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B),Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4))),Xa)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y4)
             => ? [X: B] :
                  ( member(B,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,X)),Y4) ) ) ) ) ).

% INF_le_iff
tff(fact_4471_SUP__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),C3: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C3),aa(A,B,F2,I2)) )
           => ( ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),I5)) = C3 )
            <=> ! [X: A] :
                  ( member(A,X,I5)
                 => ( aa(A,B,F2,X) = C3 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_4472_cSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),M5: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),M5) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),M5) ) ) ) ).

% cSUP_least
tff(fact_4473_cINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),M: B,F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(A,B,F2,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ).

% cINF_greatest
tff(fact_4474_INF__eq__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),C3: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),C3) )
           => ( ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I5)) = C3 )
            <=> ! [X: A] :
                  ( member(A,X,I5)
                 => ( aa(A,B,F2,X) = C3 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_4475_card__image__le,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(A),nat,finite_card(A),A4)) ) ).

% card_image_le
tff(fact_4476_SUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),B2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B2))) ) ) ) ).

% SUP_subset_mono
tff(fact_4477_INF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [B2: set(A),A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
         => ( ! [X3: A] :
                ( member(A,X3,B2)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B2))) ) ) ) ).

% INF_superset_mono
tff(fact_4478_SUP__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C3: A,A4: set(B)] :
          aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kp(A,fun(B,A),C3)),A4)) = $ite(A4 = bot_bot(set(B)),bot_bot(A),C3) ) ).

% SUP_constant
tff(fact_4479_SUP__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = bot_bot(A) ) ).

% SUP_empty
tff(fact_4480_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S: set(A),T3: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S)),T3)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_kr(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H)),T3) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),H),S) ) ) ) ) ) ).

% sum.group
tff(fact_4481_INF__inf__const1,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),Xa: B,F2: fun(A,B)] :
          ( ( I5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ks(B,fun(fun(A,B),fun(A,B)),Xa),F2)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Xa),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I5))) ) ) ) ).

% INF_inf_const1
tff(fact_4482_INF__inf__const2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [I5: set(A),F2: fun(A,B),Xa: B] :
          ( ( I5 != bot_bot(set(A)) )
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_kt(fun(A,B),fun(B,fun(A,B)),F2),Xa)),I5)) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),I5))),Xa) ) ) ) ).

% INF_inf_const2
tff(fact_4483_SUP__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: B,A4: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,F2,A3)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4))) ) ).

% SUP_insert
tff(fact_4484_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S: set(A),T3: set(B),G: fun(A,B),H: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),T3)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),S)),T3)
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ku(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S),G),H)),T3) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),H),S) ) ) ) ) ) ).

% prod.group
tff(fact_4485_INF__insert,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),A3: B,A4: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(B,A,F2,A3)),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4))) ) ).

% INF_insert
tff(fact_4486_SUP__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M5: fun(B,A),A4: set(B),B2: set(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B2))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M5),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,M5),B2))) ) ).

% SUP_union
tff(fact_4487_INF__union,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [M5: fun(B,A),A4: set(B),B2: set(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B2))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M5),A4))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,M5),B2))) ) ).

% INF_union
tff(fact_4488_INF__le__SUP,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ).

% INF_le_SUP
tff(fact_4489_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),image2(A,B,F2),A4))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B2)),aa(set(A),nat,finite_card(A),A4)) ) ) ).

% surj_card_le
tff(fact_4490_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C3: real,Xa: A,Ya: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
         => ( aa(set(A),set(A),image2(A,A,real_V8093663219630862766scaleR(A,C3)),set_or1337092689740270186AtMost(A,Xa,Ya)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C3),Xa),aa(A,A,real_V8093663219630862766scaleR(A,C3),Ya)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_4491_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H: fun(A,A),N4: set(A)] :
          ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X3)),aa(A,A,H,Y))
         => ( aa(set(A),$o,finite_finite2(A),N4)
           => ( ( N4 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic7752659483105999362nf_fin(A),N4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_4492_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H: fun(A,A),N4: set(A)] :
          ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X3)),aa(A,A,H,Y))
         => ( aa(set(A),$o,finite_finite2(A),N4)
           => ( ( N4 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic5882676163264333800up_fin(A),N4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_4493_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_4494_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_4495_lessThan__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),Na))) ).

% lessThan_Suc_eq_insert_0
tff(fact_4496_atMost__Suc__eq__insert__0,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,Na)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),Na))) ).

% atMost_Suc_eq_insert_0
tff(fact_4497_Fract__less__zero__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A3,B3)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),zero_zero(int)) ) ) ).

% Fract_less_zero_iff
tff(fact_4498_zero__less__Fract__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),fract(A3,B3))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A3) ) ) ).

% zero_less_Fract_iff
tff(fact_4499_Fract__less__one__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),fract(A3,B3)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),A3),B3) ) ) ).

% Fract_less_one_iff
tff(fact_4500_one__less__Fract__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),fract(A3,B3))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),B3),A3) ) ) ).

% one_less_Fract_iff
tff(fact_4501_Fract__le__zero__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),fract(A3,B3)),zero_zero(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),zero_zero(int)) ) ) ).

% Fract_le_zero_iff
tff(fact_4502_zero__le__Fract__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),fract(A3,B3))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A3) ) ) ).

% zero_le_Fract_iff
tff(fact_4503_Fract__le__one__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),fract(A3,B3)),one_one(rat))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),B3) ) ) ).

% Fract_le_one_iff
tff(fact_4504_one__le__Fract__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),fract(A3,B3))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B3),A3) ) ) ).

% one_le_Fract_iff
tff(fact_4505_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,Xa: A,Ya: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),times_times(A),C3)),set_or1337092689740270186AtMost(A,Xa,Ya)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Xa),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Ya)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),Ya),aa(A,A,aa(A,fun(A,A),times_times(A),C3),Xa)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_4506_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( linord4507533701916653071of_set(A,A4) = linorder_insort_key(A,A,aTP_Lamp_jn(A,A),Xa,linord4507533701916653071of_set(A,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_4507_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C3: A,Xa: A,Ya: A] :
          aa(set(A),set(A),image2(A,A,aTP_Lamp_kv(A,fun(A,A),C3)),set_or1337092689740270186AtMost(A,Xa,Ya)) = $ite(
            aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xa),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Ya),C3),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),C3))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_4508_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C3: A,A3: A,B3: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_kw(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),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),A3)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C3))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_4509_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C3: A,A3: A,B3: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_kx(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C3),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C3),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C3))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_4510_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C3: A,A3: A,B3: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_ky(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B3,M)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B3,M)),C3),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A3,M)),C3))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_4511_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C3: A,A3: A,B3: A] :
          aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),aTP_Lamp_kz(A,fun(A,fun(A,A)),M),C3)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,divide_divide(A,A3,M)),C3),aa(A,A,minus_minus(A,divide_divide(A,B3,M)),C3)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,divide_divide(A,B3,M)),C3),aa(A,A,minus_minus(A,divide_divide(A,A3,M)),C3))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_4512_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S: set(A),R: set(B),G: fun(A,B),F2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(B),$o,finite_finite2(B),R)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(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_la(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_lb(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S),G),F2)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_4513_INF__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: A,B2: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_lc(A,fun(nat,A),B2)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B2) ) ).

% INF_nat_binary
tff(fact_4514_SUP__nat__binary,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: A,B2: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_lc(A,fun(nat,A),B2)),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B2) ) ).

% SUP_nat_binary
tff(fact_4515_positive__rat,axiom,
    ! [A3: int,B3: int] :
      ( aa(rat,$o,positive,fract(A3,B3))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B3)) ) ).

% positive_rat
tff(fact_4516_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(set(nat),set(A),image2(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_4517_finite__UN,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4)))
      <=> ! [X: A] :
            ( member(A,X,A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B2,X)) ) ) ) ).

% finite_UN
tff(fact_4518_take0,axiom,
    ! [A: $tType,X2: list(A)] : take(A,zero_zero(nat),X2) = nil(A) ).

% take0
tff(fact_4519_take__eq__Nil,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( take(A,Na,Xs) = nil(A) )
    <=> ( ( Na = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_4520_take__eq__Nil2,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,Na,Xs) )
    <=> ( ( Na = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_4521_take__all,axiom,
    ! [A: $tType,Xs: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na)
     => ( take(A,Na,Xs) = Xs ) ) ).

% take_all
tff(fact_4522_take__all__iff,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( take(A,Na,Xs) = Xs )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na) ) ).

% take_all_iff
tff(fact_4523_nth__take,axiom,
    ! [A: $tType,I: nat,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),Na)
     => ( aa(nat,A,nth(A,take(A,Na,Xs)),I) = aa(nat,A,nth(A,Xs),I) ) ) ).

% nth_take
tff(fact_4524_take__update__cancel,axiom,
    ! [A: $tType,Na: nat,M: nat,Xs: list(A),Ya: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( take(A,Na,list_update(A,Xs,M,Ya)) = take(A,Na,Xs) ) ) ).

% take_update_cancel
tff(fact_4525_UN__constant,axiom,
    ! [B: $tType,A: $tType,C3: set(A),A4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ld(set(A),fun(B,set(A)),C3)),A4)) = $ite(A4 = bot_bot(set(B)),bot_bot(set(A)),C3) ).

% UN_constant
tff(fact_4526_finite__UN__I,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [A5: A] :
            ( member(A,A5,A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B2,A5)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))) ) ) ).

% finite_UN_I
tff(fact_4527_UN__Un,axiom,
    ! [A: $tType,B: $tType,M5: fun(B,set(A)),A4: set(B),B2: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B2))) = 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)),image2(B,set(A),M5),A4))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M5),B2))) ).

% UN_Un
tff(fact_4528_finite__INT,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B))] :
      ( ? [X2: A] :
          ( member(A,X2,I5)
          & aa(set(B),$o,finite_finite2(B),aa(A,set(B),A4,X2)) )
     => aa(set(B),$o,finite_finite2(B),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) ) ).

% finite_INT
tff(fact_4529_UN__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: fun(B,set(A)),C2: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_le(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B2)),C2)) = $ite(C2 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2)))) ).

% UN_simps(1)
tff(fact_4530_UN__singleton,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_lf(A,set(A))),A4)) = A4 ).

% UN_singleton
tff(fact_4531_UN__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),B2: set(A),C2: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lg(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B2)),C2)) = $ite(C2 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C2))),B2)) ).

% UN_simps(2)
tff(fact_4532_UN__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: fun(B,set(A)),C2: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lh(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),C2)) = $ite(C2 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2)))) ).

% UN_simps(3)
tff(fact_4533_UN__insert,axiom,
    ! [A: $tType,B: $tType,B2: fun(B,set(A)),A3: B,A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),B2,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),A4))) ).

% UN_insert
tff(fact_4534_INT__insert,axiom,
    ! [A: $tType,B: $tType,B2: fun(B,set(A)),A3: B,A4: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),A4))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),B2,A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),A4))) ).

% INT_insert
tff(fact_4535_Sup__SUP__eq,axiom,
    ! [A: $tType,S: set(fun(A,$o)),X2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),S),X2)
    <=> member(A,X2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(fun(A,$o)),set(set(A)),image2(fun(A,$o),set(A),collect(A)),S))) ) ).

% Sup_SUP_eq
tff(fact_4536_SUP__Sup__eq,axiom,
    ! [A: $tType,S: set(set(A)),X2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image2(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o))),S)),X2)
    <=> member(A,X2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S)) ) ).

% SUP_Sup_eq
tff(fact_4537_SUP__UN__eq,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,set(A)),S: set(B),X2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Sup_Sup(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),aTP_Lamp_li(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S)),X2)
    <=> member(A,X2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R2),S))) ) ).

% SUP_UN_eq
tff(fact_4538_SUP__Sup__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S)) ) ).

% SUP_Sup_eq2
tff(fact_4539_SUP__UN__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S: set(C),X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_lj(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S)),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R2),S))) ) ).

% SUP_UN_eq2
tff(fact_4540_Inf__INT__eq,axiom,
    ! [A: $tType,S: set(fun(A,$o)),X2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),S),X2)
    <=> member(A,X2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(fun(A,$o)),set(set(A)),image2(fun(A,$o),set(A),collect(A)),S))) ) ).

% Inf_INT_eq
tff(fact_4541_INF__Int__eq,axiom,
    ! [A: $tType,S: set(set(A)),X2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(set(A)),set(fun(A,$o)),image2(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o))),S)),X2)
    <=> member(A,X2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S)) ) ).

% INF_Int_eq
tff(fact_4542_INF__INT__eq,axiom,
    ! [A: $tType,B: $tType,R2: fun(B,set(A)),S: set(B),X2: A] :
      ( aa(A,$o,aa(set(fun(A,$o)),fun(A,$o),complete_Inf_Inf(fun(A,$o)),aa(set(B),set(fun(A,$o)),image2(B,fun(A,$o),aTP_Lamp_li(fun(B,set(A)),fun(B,fun(A,$o)),R2)),S)),X2)
    <=> member(A,X2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),R2),S))) ) ).

% INF_INT_eq
tff(fact_4543_INF__INT__eq2,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: fun(C,set(product_prod(A,B))),S: set(C),X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(C),set(fun(A,fun(B,$o))),image2(C,fun(A,fun(B,$o)),aTP_Lamp_lj(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),R2)),S)),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),R2),S))) ) ).

% INF_INT_eq2
tff(fact_4544_INF__Int__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B))),X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),aa(set(set(product_prod(A,B))),set(fun(A,fun(B,$o))),image2(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o)))),S)),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),S)) ) ).

% INF_Int_eq2
tff(fact_4545_Sup__SUP__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Sup_Sup(fun(A,fun(B,$o))),S),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).

% Sup_SUP_eq2
tff(fact_4546_Inf__INT__eq2,axiom,
    ! [B: $tType,A: $tType,S: set(fun(A,fun(B,$o))),X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(fun(A,fun(B,$o))),fun(A,fun(B,$o)),complete_Inf_Inf(fun(A,fun(B,$o))),S),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(fun(product_prod(A,B),$o)),set(set(product_prod(A,B))),image2(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B))),aa(set(fun(A,fun(B,$o))),set(fun(product_prod(A,B),$o)),image2(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o)),S)))) ) ).

% Inf_INT_eq2
tff(fact_4547_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B2: fun(B,set(A)),A4: set(B)] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image2(B,set(set(A)),aTP_Lamp_lk(fun(B,set(A)),fun(B,set(set(A))),B2)),A4))),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),A4)))) ).

% UN_Pow_subset
tff(fact_4548_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_4549_set__take__subset,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,Na,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_take_subset
tff(fact_4550_less__rat__def,axiom,
    ! [Xa: rat,Ya: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Xa),Ya)
    <=> aa(rat,$o,positive,aa(rat,rat,minus_minus(rat,Ya),Xa)) ) ).

% less_rat_def
tff(fact_4551_INF__filter__not__bot,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F3: fun(A,filter(B))] :
      ( ! [X6: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),B2)
         => ( aa(set(A),$o,finite_finite2(A),X6)
           => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F3),X6)) != bot_bot(filter(B)) ) ) )
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F3),B2)) != bot_bot(filter(B)) ) ) ).

% INF_filter_not_bot
tff(fact_4552_finite__int__iff__bounded__le,axiom,
    ! [S: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S)),aa(int,set(int),set_ord_atMost(int),K3)) ) ).

% finite_int_iff_bounded_le
tff(fact_4553_finite__int__iff__bounded,axiom,
    ! [S: set(int)] :
      ( aa(set(int),$o,finite_finite2(int),S)
    <=> ? [K3: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image2(int,int,abs_abs(int)),S)),aa(int,set(int),set_ord_lessThan(int),K3)) ) ).

% finite_int_iff_bounded
tff(fact_4554_UNION__empty__conv_I2_J,axiom,
    ! [B: $tType,A: $tType,B2: fun(B,set(A)),A4: set(B)] :
      ( ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),A4)) = bot_bot(set(A)) )
    <=> ! [X: B] :
          ( member(B,X,A4)
         => ( aa(B,set(A),B2,X) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(2)
tff(fact_4555_UNION__empty__conv_I1_J,axiom,
    ! [B: $tType,A: $tType,B2: fun(B,set(A)),A4: set(B)] :
      ( ( bot_bot(set(A)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),A4)) )
    <=> ! [X: B] :
          ( member(B,X,A4)
         => ( aa(B,set(A),B2,X) = bot_bot(set(A)) ) ) ) ).

% UNION_empty_conv(1)
tff(fact_4556_UN__empty,axiom,
    ! [B: $tType,A: $tType,B2: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),bot_bot(set(B)))) = bot_bot(set(A)) ).

% UN_empty
tff(fact_4557_UN__empty2,axiom,
    ! [B: $tType,A: $tType,A4: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ll(B,set(A))),A4)) = bot_bot(set(A)) ).

% UN_empty2
tff(fact_4558_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),I5: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5))),B2)
    <=> ! [X: B] :
          ( member(B,X,I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),A4,X)),B2) ) ) ).

% UN_subset_iff
tff(fact_4559_UN__upper,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B2: fun(A,set(B))] :
      ( member(A,A3,A4)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B2,A3)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))) ) ).

% UN_upper
tff(fact_4560_UN__least,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: fun(A,set(B)),C2: set(B)] :
      ( ! [X3: A] :
          ( member(A,X3,A4)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B2,X3)),C2) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))),C2) ) ).

% UN_least
tff(fact_4561_UN__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( ! [X3: A] :
            ( member(A,X3,A4)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,X3)),aa(A,set(B),G,X3)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),A4))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),B2))) ) ) ).

% UN_mono
tff(fact_4562_UN__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A4: set(A),A3: B,B2: fun(A,set(B))] :
      ( member(A,U,A4)
     => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_lm(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B2)),A4)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))) ) ) ).

% UN_insert_distrib
tff(fact_4563_Un__Union__image,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),B2: fun(B,set(A)),C2: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ln(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),C2)) = 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)),image2(B,set(A),A4),C2))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2))) ).

% Un_Union_image
tff(fact_4564_UN__Un__distrib,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),B2: fun(B,set(A)),I5: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ln(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),I5)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),I5))) ).

% UN_Un_distrib
tff(fact_4565_UN__absorb,axiom,
    ! [B: $tType,A: $tType,K: A,I5: set(A),A4: fun(A,set(B))] :
      ( member(A,K,I5)
     => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),A4,K)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) = aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5)) ) ) ).

% UN_absorb
tff(fact_4566_INT__lower,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B2: fun(A,set(B))] :
      ( member(A,A3,A4)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))),aa(A,set(B),B2,A3)) ) ).

% INT_lower
tff(fact_4567_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A4: set(A),C2: set(B),B2: fun(A,set(B))] :
      ( ! [X3: A] :
          ( member(A,X3,A4)
         => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C2),aa(A,set(B),B2,X3)) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))) ) ).

% INT_greatest
tff(fact_4568_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(A),F2: fun(A,set(B)),G: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( ! [X3: A] :
            ( member(A,X3,A4)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,X3)),aa(A,set(B),G,X3)) )
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),B2))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),G),A4))) ) ) ).

% INT_anti_mono
tff(fact_4569_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B2: set(A),A4: fun(B,set(A)),I5: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5)))
    <=> ! [X: B] :
          ( member(B,X,I5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(B,set(A),A4,X)) ) ) ).

% INT_subset_iff
tff(fact_4570_INT__extend__simps_I5_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: fun(B,set(A)),C2: set(B)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_le(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B2)),C2)) ).

% INT_extend_simps(5)
tff(fact_4571_INT__insert__distrib,axiom,
    ! [B: $tType,A: $tType,U: A,A4: set(A),A3: B,B2: fun(A,set(B))] :
      ( member(A,U,A4)
     => ( aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_lm(B,fun(fun(A,set(B)),fun(A,set(B))),A3),B2)),A4)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))) ) ) ).

% INT_insert_distrib
tff(fact_4572_INT__extend__simps_I7_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: fun(B,set(A)),C2: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lh(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),C2)) ).

% INT_extend_simps(7)
tff(fact_4573_INT__extend__simps_I6_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C2: set(B),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C2))),B2) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lg(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B2)),C2)) ).

% INT_extend_simps(6)
tff(fact_4574_Un__INT__distrib,axiom,
    ! [A: $tType,B: $tType,B2: set(A),A4: fun(B,set(A)),I5: set(B)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lh(set(A),fun(fun(B,set(A)),fun(B,set(A))),B2),A4)),I5)) ).

% Un_INT_distrib
tff(fact_4575_Un__INT__distrib2,axiom,
    ! [C: $tType,A: $tType,B: $tType,A4: fun(B,set(A)),I5: set(B),B2: fun(C,set(A)),J4: set(C)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B2),J4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_lp(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),A4),B2),J4)),I5)) ).

% Un_INT_distrib2
tff(fact_4576_Un__Inter,axiom,
    ! [A: $tType,A4: set(A),B2: set(set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B2)) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4)),B2)) ).

% Un_Inter
tff(fact_4577_set__take__subset__set__take,axiom,
    ! [A: $tType,M: nat,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,M,Xs))),aa(list(A),set(A),set2(A),take(A,Na,Xs))) ) ).

% set_take_subset_set_take
tff(fact_4578_in__image__insert__iff,axiom,
    ! [A: $tType,B2: set(set(A)),Xa: A,A4: set(A)] :
      ( ! [C7: set(A)] :
          ( member(set(A),C7,B2)
         => ~ member(A,Xa,C7) )
     => ( member(set(A),A4,aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa)),B2))
      <=> ( member(A,Xa,A4)
          & member(set(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))),B2) ) ) ) ).

% in_image_insert_iff
tff(fact_4579_UN__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A3: A,B2: fun(B,set(A)),C2: set(B)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2))) = $ite(C2 = bot_bot(set(B)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_le(A,fun(fun(B,set(A)),fun(B,set(A))),A3),B2)),C2))) ).

% UN_extend_simps(1)
tff(fact_4580_UN__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: fun(B,set(A)),C2: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2))) = $ite(C2 = bot_bot(set(B)),A4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lh(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),C2))) ).

% UN_extend_simps(3)
tff(fact_4581_UN__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C2: set(B),B2: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C2))),B2) = $ite(C2 = bot_bot(set(B)),B2,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lg(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B2)),C2))) ).

% UN_extend_simps(2)
tff(fact_4582_INT__extend__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C2: set(B),B2: set(A)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C2))),B2) = $ite(C2 = bot_bot(set(B)),B2,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lq(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B2)),C2))) ).

% INT_extend_simps(1)
tff(fact_4583_INT__extend__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: fun(B,set(A)),C2: set(B)] :
      aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2))) = $ite(C2 = bot_bot(set(B)),A4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lr(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),C2))) ).

% INT_extend_simps(2)
tff(fact_4584_Pow__insert,axiom,
    ! [A: $tType,A3: A,A4: set(A)] : pow2(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = 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,A4)),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3)),pow2(A,A4))) ).

% Pow_insert
tff(fact_4585_INT__Un,axiom,
    ! [A: $tType,B: $tType,M5: fun(B,set(A)),A4: set(B),B2: set(B)] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M5),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M5),A4))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),M5),B2))) ).

% INT_Un
tff(fact_4586_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys2: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys2))
       => ( ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),K)
             => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys2),I2) ) )
         => ( take(A,K,Xs) = take(A,K,Ys2) ) ) ) ) ).

% nth_take_lemma
tff(fact_4587_INT__extend__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: fun(B,set(A)),C2: set(B)] :
      aa(set(A),set(A),minus_minus(set(A),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2))) = $ite(C2 = bot_bot(set(B)),A4,aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ls(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),C2))) ).

% INT_extend_simps(4)
tff(fact_4588_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),U)
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image2(nat,int,semiring_1_of_nat(int)),aa(nat,set(nat),set_ord_lessThan(nat),aa(int,nat,nat2,U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_4589_sum_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [I5: set(A),A4: fun(A,set(B)),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [X3: A] :
                ( member(A,X3,I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A4,X3)) )
           => ( ! [X3: A] :
                  ( member(A,X3,I5)
                 => ! [Xa4: A] :
                      ( member(A,Xa4,I5)
                     => ( ( X3 != Xa4 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,X3)),aa(A,set(B),A4,Xa4)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) = 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_lt(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A4),G)),I5) ) ) ) ) ) ).

% sum.UNION_disjoint
tff(fact_4590_prod_OUNION__disjoint,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [I5: set(A),A4: fun(A,set(B)),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [X3: A] :
                ( member(A,X3,I5)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A4,X3)) )
           => ( ! [X3: A] :
                  ( member(A,X3,I5)
                 => ! [Xa4: A] :
                      ( member(A,Xa4,I5)
                     => ( ( X3 != Xa4 )
                       => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,X3)),aa(A,set(B),A4,Xa4)) = bot_bot(set(B)) ) ) ) )
             => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_lu(fun(A,set(B)),fun(fun(B,C),fun(A,C)),A4),G)),I5) ) ) ) ) ) ).

% prod.UNION_disjoint
tff(fact_4591_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_lv(fun(A,set(B)),fun(A,nat),A4)),I5)) ) ).

% card_UN_le
tff(fact_4592_card__UN__disjoint,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),I5)
     => ( ! [X3: A] :
            ( member(A,X3,I5)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),A4,X3)) )
       => ( ! [X3: A] :
              ( member(A,X3,I5)
             => ! [Xa4: A] :
                  ( member(A,Xa4,I5)
                 => ( ( X3 != Xa4 )
                   => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(A,set(B),A4,X3)),aa(A,set(B),A4,Xa4)) = bot_bot(set(B)) ) ) ) )
         => ( aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_lv(fun(A,set(B)),fun(A,nat),A4)),I5) ) ) ) ) ).

% card_UN_disjoint
tff(fact_4593_UN__le__eq__Un0,axiom,
    ! [A: $tType,M5: fun(nat,set(A)),Na: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),M5),aa(nat,set(nat),set_ord_atMost(nat),Na))) = 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)),image2(nat,set(A),M5),set_or1337092689740270186AtMost(nat,one_one(nat),Na)))),aa(nat,set(A),M5,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_4594_Union__image__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,set(A)),A3: B,B2: set(B)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),B2))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),F2,A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),B2))) ).

% Union_image_insert
tff(fact_4595_Union__image__empty,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(B,set(A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),bot_bot(set(B))))) = A4 ).

% Union_image_empty
tff(fact_4596_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(B,set(A)),G: fun(C,set(B)),Xa: C,X4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),aa(C,set(B),G,Xa)))),X4)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(C,set(B),G,Xa)),aa(fun(B,$o),set(B),collect(B),aa(set(A),fun(B,$o),aTP_Lamp_lw(fun(B,set(A)),fun(set(A),fun(B,$o)),F2),X4))) ) ).

% UN_image_subset
tff(fact_4597_Pow__fold,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( pow2(A,A4) = finite_fold(A,set(set(A)),aTP_Lamp_lx(A,fun(set(set(A)),set(set(A)))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))),A4) ) ) ).

% Pow_fold
tff(fact_4598_fold__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Z: A] : finite_fold(B,A,F2,Z,bot_bot(set(B))) = Z ).

% fold_empty
tff(fact_4599_fold__infinite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( finite_fold(A,B,F2,Z,A4) = Z ) ) ).

% fold_infinite
tff(fact_4600_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F3: fun(A,filter(B))] :
      ( ! [I2: A] :
          ( member(A,I2,I5)
         => ! [J2: A] :
              ( member(A,J2,I5)
             => ? [X2: A] :
                  ( member(A,X2,I5)
                  & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F3,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F3,I2)),aa(A,filter(B),F3,J2))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F3),I5)) = bot_bot(filter(B)) )
      <=> ? [X: A] :
            ( member(A,X,I5)
            & ( aa(A,filter(B),F3,X) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_4601_fold__closed__eq,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),Z: B] :
      ( ! [A5: A,B5: B] :
          ( member(A,A5,A4)
         => ( member(B,B5,B2)
           => ( aa(B,B,aa(A,fun(B,B),F2,A5),B5) = aa(B,B,aa(A,fun(B,B),G,A5),B5) ) ) )
     => ( ! [A5: A,B5: B] :
            ( member(A,A5,A4)
           => ( member(B,B5,B2)
             => member(B,aa(B,B,aa(A,fun(B,B),G,A5),B5),B2) ) )
       => ( member(B,Z,B2)
         => ( finite_fold(A,B,F2,Z,A4) = finite_fold(A,B,G,Z,A4) ) ) ) ) ).

% fold_closed_eq
tff(fact_4602_Inf__filter__not__bot,axiom,
    ! [A: $tType,B2: set(filter(A))] :
      ( ! [X6: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X6),B2)
         => ( aa(set(filter(A)),$o,finite_finite2(filter(A)),X6)
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X6) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B2) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_4603_union__fold__insert,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = finite_fold(A,set(A),insert2(A),B2,A4) ) ) ).

% union_fold_insert
tff(fact_4604_sup__Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),B2) = finite_fold(A,A,sup_sup(A),B2,A4) ) ) ) ).

% sup_Sup_fold_sup
tff(fact_4605_inf__Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A),B2: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),B2) = finite_fold(A,A,inf_inf(A),B2,A4) ) ) ) ).

% inf_Inf_fold_inf
tff(fact_4606_minus__fold__remove,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),set(A),minus_minus(set(A),B2),A4) = finite_fold(A,set(A),remove(A),B2,A4) ) ) ).

% minus_fold_remove
tff(fact_4607_Sup__fold__sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,complete_Sup_Sup(A),A4) = finite_fold(A,A,sup_sup(A),bot_bot(A),A4) ) ) ) ).

% Sup_fold_sup
tff(fact_4608_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = finite_fold(A,A,inf_inf(A),Xa,A4) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_4609_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = finite_fold(A,A,sup_sup(A),Xa,A4) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_4610_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),set(B),image2(A,B,F2),A4) = finite_fold(A,set(B),aTP_Lamp_ly(fun(A,B),fun(A,fun(set(B),set(B))),F2),bot_bot(set(B)),A4) ) ) ).

% image_fold_insert
tff(fact_4611_conj__subset__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P))
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(fun(A,$o),set(A),collect(A),Q)) ) ) ).

% conj_subset_def
tff(fact_4612_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F2: fun(nat,set(A)),S: set(A)] :
      ( ! [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F2,I2)),S)
     => ( aa(set(A),$o,finite_finite2(A),S)
       => ( ? [N7: nat] :
              ( ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N7)
                 => ! [M4: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N7)
                     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N2)
                       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),aa(nat,set(A),F2,M4)),aa(nat,set(A),F2,N2)) ) ) )
              & ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
                 => ( aa(nat,set(A),F2,N7) = 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)),image2(nat,set(A),F2),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_4613_Set__filter__fold,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( filter3(A,P,A4) = finite_fold(A,set(A),aTP_Lamp_lz(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A4) ) ) ).

% Set_filter_fold
tff(fact_4614_suminf__eq__SUP__real,axiom,
    ! [X4: fun(nat,real)] :
      ( summable(real,X4)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,X4,I2))
       => ( suminf(real,X4) = aa(set(real),real,complete_Sup_Sup(real),aa(set(nat),set(real),image2(nat,real,aTP_Lamp_ma(fun(nat,real),fun(nat,real),X4)),top_top(set(nat)))) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_4615_top__apply,axiom,
    ! [B: $tType,A: $tType] :
      ( top(A)
     => ! [Xa: B] : aa(B,A,top_top(fun(B,A)),Xa) = top_top(A) ) ).

% top_apply
tff(fact_4616_UNIV__I,axiom,
    ! [A: $tType,Xa: A] : member(A,Xa,top_top(set(A))) ).

% UNIV_I
tff(fact_4617_member__filter,axiom,
    ! [A: $tType,Xa: A,P: fun(A,$o),A4: set(A)] :
      ( member(A,Xa,filter3(A,P,A4))
    <=> ( member(A,Xa,A4)
        & aa(A,$o,P,Xa) ) ) ).

% member_filter
tff(fact_4618_finite__Plus__UNIV__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(sum_sum(A,B)),$o,finite_finite2(sum_sum(A,B)),top_top(set(sum_sum(A,B))))
    <=> ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
        & aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_Plus_UNIV_iff
tff(fact_4619_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),top_top(set(option(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_4620_inf__top__left,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),Xa) = Xa ) ).

% inf_top_left
tff(fact_4621_inf__top__right,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),top_top(A)) = Xa ) ).

% inf_top_right
tff(fact_4622_inf__eq__top__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) = top_top(A) )
        <=> ( ( Xa = top_top(A) )
            & ( Ya = top_top(A) ) ) ) ) ).

% inf_eq_top_iff
tff(fact_4623_top__eq__inf__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [Xa: A,Ya: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) )
        <=> ( ( Xa = top_top(A) )
            & ( Ya = top_top(A) ) ) ) ) ).

% top_eq_inf_iff
tff(fact_4624_inf__top_Oeq__neutr__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) = top_top(A) )
        <=> ( ( A3 = top_top(A) )
            & ( B3 = top_top(A) ) ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_4625_inf__top_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),top_top(A)),A3) = A3 ) ).

% inf_top.left_neutral
tff(fact_4626_inf__top_Oneutr__eq__iff,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A,B3: A] :
          ( ( top_top(A) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) )
        <=> ( ( A3 = top_top(A) )
            & ( B3 = top_top(A) ) ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_4627_inf__top_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bounde4346867609351753570nf_top(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),top_top(A)) = A3 ) ).

% inf_top.right_neutral
tff(fact_4628_boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),Xa) = top_top(A) ) ).

% boolean_algebra.disj_one_left
tff(fact_4629_boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),top_top(A)) = top_top(A) ) ).

% boolean_algebra.disj_one_right
tff(fact_4630_sup__top__left,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),top_top(A)),Xa) = top_top(A) ) ).

% sup_top_left
tff(fact_4631_sup__top__right,axiom,
    ! [A: $tType] :
      ( bounded_lattice_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),top_top(A)) = top_top(A) ) ).

% sup_top_right
tff(fact_4632_Int__UNIV,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = top_top(set(A)) )
    <=> ( ( A4 = top_top(set(A)) )
        & ( B2 = top_top(set(A)) ) ) ) ).

% Int_UNIV
tff(fact_4633_max__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),top_top(A)) = top_top(A) ) ).

% max_top2
tff(fact_4634_max__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),top_top(A)),Xa) = top_top(A) ) ).

% max_top
tff(fact_4635_Pow__UNIV,axiom,
    ! [A: $tType] : pow2(A,top_top(set(A))) = top_top(set(set(A))) ).

% Pow_UNIV
tff(fact_4636_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mb($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_4637_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_4638_Sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),A4) = top_top(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
             => ? [Xa3: A] :
                  ( member(A,Xa3,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa3) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_4639_boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),bot_bot(A)) = top_top(A) ) ) ).

% boolean_algebra.compl_zero
tff(fact_4640_boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ( aa(A,A,uminus_uminus(A),top_top(A)) = bot_bot(A) ) ) ).

% boolean_algebra.compl_one
tff(fact_4641_boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,uminus_uminus(A),Xa)) = top_top(A) ) ).

% boolean_algebra.disj_cancel_right
tff(fact_4642_boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),Xa) = top_top(A) ) ).

% boolean_algebra.disj_cancel_left
tff(fact_4643_sup__compl__top__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),Ya)) = top_top(A) ) ).

% sup_compl_top_left2
tff(fact_4644_sup__compl__top__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya)) = top_top(A) ) ).

% sup_compl_top_left1
tff(fact_4645_Inf__UNIV,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) = bot_bot(A) ) ) ).

% Inf_UNIV
tff(fact_4646_ccInf__empty,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = top_top(A) ) ) ).

% ccInf_empty
tff(fact_4647_Inf__empty,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = top_top(A) ) ) ).

% Inf_empty
tff(fact_4648_Diff__UNIV,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_4649_surj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Na: nat] :
      ( ( aa(set(A),set(A),image2(A,A,F2),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image2(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F2)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_4650_finite__compl,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),uminus_uminus(set(A)),A4))
      <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_compl
tff(fact_4651_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F2: fun(B,A),A4: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4)) = top_top(A) )
        <=> ! [X: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),top_top(A))
             => ? [Xa3: B] :
                  ( member(B,Xa3,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),aa(B,A,F2,Xa3)) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_4652_range__constant,axiom,
    ! [B: $tType,A: $tType,Xa: A] : aa(set(B),set(A),image2(B,A,aTP_Lamp_kf(A,fun(B,A),Xa)),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ).

% range_constant
tff(fact_4653_ccINF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).

% ccINF_empty
tff(fact_4654_INT__constant,axiom,
    ! [B: $tType,A: $tType,C3: set(A),A4: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_ld(set(A),fun(B,set(A)),C3)),A4)) = $ite(A4 = bot_bot(set(B)),top_top(set(A)),C3) ).

% INT_constant
tff(fact_4655_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),Xa)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_4656_INT__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),B2: set(A),C2: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lq(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B2)),C2)) = $ite(C2 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C2))),B2)) ).

% INT_simps(1)
tff(fact_4657_INT__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: fun(B,set(A)),C2: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lr(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),C2)) = $ite(C2 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2)))) ).

% INT_simps(2)
tff(fact_4658_INT__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),B2: set(A),C2: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_mc(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B2)),C2)) = $ite(C2 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),minus_minus(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C2))),B2)) ).

% INT_simps(3)
tff(fact_4659_INT__simps_I4_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: fun(B,set(A)),C2: set(B)] :
      aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ls(set(A),fun(fun(B,set(A)),fun(B,set(A))),A4),B2)),C2)) = $ite(C2 = bot_bot(set(B)),top_top(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),C2)))) ).

% INT_simps(4)
tff(fact_4660_range__eqI,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A),Xa: B] :
      ( ( B3 = aa(B,A,F2,Xa) )
     => member(A,B3,aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ) ).

% range_eqI
tff(fact_4661_rangeI,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xa: B] : member(A,aa(B,A,F2,Xa),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ).

% rangeI
tff(fact_4662_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image2(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_md(fun(C,A),fun(fun(B,C),fun(B,A)),F2),G)),top_top(set(B))) = aa(set(C),set(A),image2(C,A,F2),aa(set(B),set(C),image2(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_4663_rangeE,axiom,
    ! [A: $tType,B: $tType,B3: A,F2: fun(B,A)] :
      ( member(A,B3,aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
     => ~ ! [X3: B] : B3 != aa(B,A,F2,X3) ) ).

% rangeE
tff(fact_4664_less__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less(filter(A)),F3),F8)
    <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F8)
        & ~ aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F8),F3) ) ) ).

% less_filter_def
tff(fact_4665_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [Xa: A,Ya: A] :
          ( ( set_or1337092689740270186AtMost(A,Xa,Ya) = top_top(set(A)) )
        <=> ( ( Xa = bot_bot(A) )
            & ( Ya = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_4666_empty__not__UNIV,axiom,
    ! [A: $tType] : bot_bot(set(A)) != top_top(set(A)) ).

% empty_not_UNIV
tff(fact_4667_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),top_top(A)) ) ).

% top_greatest
tff(fact_4668_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A3)
        <=> ( A3 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_4669_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A3)
         => ( A3 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_4670_subset__UNIV,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),top_top(set(A))) ).

% subset_UNIV
tff(fact_4671_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),top_top(A)),A3) ) ).

% top.extremum_strict
tff(fact_4672_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_4673_infinite__UNIV__char__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% infinite_UNIV_char_0
tff(fact_4674_ex__new__if__finite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ? [A5: A] : ~ member(A,A5,A4) ) ) ).

% ex_new_if_finite
tff(fact_4675_finite__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite(A)
     => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_UNIV
tff(fact_4676_finite__fun__UNIVD2,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ).

% finite_fun_UNIVD2
tff(fact_4677_Finite__Set_Ofinite__set,axiom,
    ! [A: $tType] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),top_top(set(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% Finite_Set.finite_set
tff(fact_4678_finite__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B))))
    <=> ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
        & aa(set(B),$o,finite_finite2(B),top_top(set(B))) ) ) ).

% finite_prod
tff(fact_4679_finite__Prod__UNIV,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(set(B),$o,finite_finite2(B),top_top(set(B)))
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),top_top(set(product_prod(A,B)))) ) ) ).

% finite_Prod_UNIV
tff(fact_4680_UNIV__eq__I,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ! [X3: A] : member(A,X3,A4)
     => ( top_top(set(A)) = A4 ) ) ).

% UNIV_eq_I
tff(fact_4681_UNIV__witness,axiom,
    ! [A: $tType] :
    ? [X3: A] : member(A,X3,top_top(set(A))) ).

% UNIV_witness
tff(fact_4682_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A4: set(A)] : filter3(A,P,A4) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_me(fun(A,$o),fun(set(A),fun(A,$o)),P),A4)) ).

% Set.filter_def
tff(fact_4683_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_mf(A,$o)) ).

% UNIV_def
tff(fact_4684_nat__not__finite,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% nat_not_finite
tff(fact_4685_infinite__UNIV__nat,axiom,
    ~ aa(set(nat),$o,finite_finite2(nat),top_top(set(nat))) ).

% infinite_UNIV_nat
tff(fact_4686_insert__UNIV,axiom,
    ! [A: $tType,Xa: A] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),top_top(set(A))) = top_top(set(A)) ).

% insert_UNIV
tff(fact_4687_Un__UNIV__left,axiom,
    ! [A: $tType,B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),top_top(set(A))),B2) = top_top(set(A)) ).

% Un_UNIV_left
tff(fact_4688_Un__UNIV__right,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),top_top(set(A))) = top_top(set(A)) ).

% Un_UNIV_right
tff(fact_4689_Int__UNIV__right,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),top_top(set(A))) = A4 ).

% Int_UNIV_right
tff(fact_4690_Int__UNIV__left,axiom,
    ! [A: $tType,B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),top_top(set(A))),B2) = B2 ).

% Int_UNIV_left
tff(fact_4691_boolean__algebra_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),top_top(A)) = Xa ) ).

% boolean_algebra.conj_one_right
tff(fact_4692_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),top_top(set(fun(A,B))))
     => ( ( aa(set(B),nat,finite_card(B),top_top(set(B))) != aa(nat,nat,suc,zero_zero(nat)) )
       => aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% finite_fun_UNIVD1
tff(fact_4693_sup__cancel__left2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),B3)) = top_top(A) ) ).

% sup_cancel_left2
tff(fact_4694_sup__cancel__left1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),A3)),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Xa)),B3)) = top_top(A) ) ).

% sup_cancel_left1
tff(fact_4695_Inf__sup__eq__top__iff,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [B2: set(A),A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B2)),A3) = top_top(A) )
        <=> ! [X: A] :
              ( member(A,X,B2)
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),A3) = top_top(A) ) ) ) ) ).

% Inf_sup_eq_top_iff
tff(fact_4696_finite__filter,axiom,
    ! [A: $tType,S: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => aa(set(A),$o,finite_finite2(A),filter3(A,P,S)) ) ).

% finite_filter
tff(fact_4697_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),B2: set(A),I: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))),B2)
     => member(A,aa(B,A,F2,I),B2) ) ).

% range_subsetD
tff(fact_4698_perfect__space__class_OUNIV__not__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xa: A] : top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ) ).

% perfect_space_class.UNIV_not_singleton
tff(fact_4699_bot__finite__def,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( bot_bot(A) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).

% bot_finite_def
tff(fact_4700_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H)) ) ).

% not_UNIV_le_Icc
tff(fact_4701_card__eq__UNIV__imp__eq__UNIV,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),nat,finite_card(A),A4) = aa(set(A),nat,finite_card(A),top_top(set(A))) )
       => ( A4 = top_top(set(A)) ) ) ) ).

% card_eq_UNIV_imp_eq_UNIV
tff(fact_4702_finite__range__Some,axiom,
    ! [A: $tType] :
      ( aa(set(option(A)),$o,finite_finite2(option(A)),aa(set(A),set(option(A)),image2(A,option(A),some(A)),top_top(set(A))))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% finite_range_Some
tff(fact_4703_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H)) ) ).

% not_UNIV_le_Iic
tff(fact_4704_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_4705_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_4706_Compl__partition2,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A4)),A4) = top_top(set(A)) ).

% Compl_partition2
tff(fact_4707_Compl__partition,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),uminus_uminus(set(A)),A4)) = top_top(set(A)) ).

% Compl_partition
tff(fact_4708_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),A4) ).

% Compl_eq_Diff_UNIV
tff(fact_4709_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F2: fun(A,C)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,G),top_top(set(B))))
     => aa(set(C),$o,finite_finite2(C),aa(set(B),set(C),image2(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_mg(fun(B,A),fun(fun(A,C),fun(B,C)),G),F2)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_4710_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) = top_top(A) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xa)),Ya) ) ) ).

% sup_shunt
tff(fact_4711_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [A3: A,Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Xa) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Xa) = top_top(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),Ya) = bot_bot(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),Ya) = top_top(A) )
               => ( Xa = Ya ) ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_4712_range__eq__singletonD,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: A,Xa: B] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))) )
     => ( aa(B,A,F2,Xa) = A3 ) ) ).

% range_eq_singletonD
tff(fact_4713_INF__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),bot_bot(set(B)))) = top_top(A) ) ).

% INF_empty
tff(fact_4714_INF__constant,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [C3: A,A4: set(B)] :
          aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,aTP_Lamp_kp(A,fun(B,A),C3)),A4)) = $ite(A4 = bot_bot(set(B)),top_top(A),C3) ) ).

% INF_constant
tff(fact_4715_Sup__finite__empty,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( aa(set(A),A,complete_Sup_Sup(A),bot_bot(set(A))) = aa(set(A),A,complete_Inf_Inf(A),top_top(set(A))) ) ) ).

% Sup_finite_empty
tff(fact_4716_Inf__finite__empty,axiom,
    ! [A: $tType] :
      ( finite_lattice(A)
     => ( aa(set(A),A,complete_Inf_Inf(A),bot_bot(set(A))) = aa(set(A),A,complete_Sup_Sup(A),top_top(set(A))) ) ) ).

% Inf_finite_empty
tff(fact_4717_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image2(B,A,F2),A4))),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),uminus_uminus(set(B)),A4))) ) ).

% surj_Compl_image_subset
tff(fact_4718_card_Oeq__fold,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),nat,finite_card(A),A4) = finite_fold(A,nat,aTP_Lamp_mh(A,fun(nat,nat)),zero_zero(nat),A4) ).

% card.eq_fold
tff(fact_4719_INT__empty,axiom,
    ! [B: $tType,A: $tType,B2: fun(B,set(A))] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),bot_bot(set(B)))) = top_top(set(A)) ).

% INT_empty
tff(fact_4720_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),Ya) = bot_bot(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),Ya) = top_top(A) )
           => ( aa(A,A,uminus_uminus(A),Xa) = Ya ) ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_4721_Inf__fold__inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,complete_Inf_Inf(A),A4) = finite_fold(A,A,inf_inf(A),top_top(A),A4) ) ) ) ).

% Inf_fold_inf
tff(fact_4722_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_4723_range__enumerate,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => ( aa(set(nat),set(nat),image2(nat,nat,infini527867602293511546merate(nat,S)),top_top(set(nat))) = S ) ) ).

% range_enumerate
tff(fact_4724_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A)))) ) ).

% finite_UNIV_card_ge_0
tff(fact_4725_inter__Set__filter,axiom,
    ! [A: $tType,B2: set(A),A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = filter3(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4),B2) ) ) ).

% inter_Set_filter
tff(fact_4726_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),aa(set(nat),set(nat),image2(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_4727_INT__extend__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),C2: set(B),B2: set(A)] :
      aa(set(A),set(A),minus_minus(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),C2))),B2) = $ite(C2 = bot_bot(set(B)),aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),B2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_mc(fun(B,set(A)),fun(set(A),fun(B,set(A))),A4),B2)),C2))) ).

% INT_extend_simps(3)
tff(fact_4728_UN__UN__finite__eq,axiom,
    ! [A: $tType,A4: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),aTP_Lamp_mi(fun(nat,set(A)),fun(nat,set(A)),A4)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_4729_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))) ) ).

% card_range_greater_zero
tff(fact_4730_UN__finite__subset,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),C2: set(A)] :
      ( ! [N2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),C2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat))))),C2) ) ).

% UN_finite_subset
tff(fact_4731_UN__finite2__eq,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),B2: 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)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B2),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)),image2(nat,set(A),A4),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B2),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_4732_range__mod,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(set(nat),set(nat),image2(nat,nat,aTP_Lamp_mj(nat,fun(nat,nat),Na)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Na) ) ) ).

% range_mod
tff(fact_4733_fold__union__pair,axiom,
    ! [B: $tType,A: $tType,B2: set(A),Xa: B,A4: set(product_prod(B,A))] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => ( aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(A),set(set(product_prod(B,A))),image2(A,set(product_prod(B,A)),aTP_Lamp_mk(B,fun(A,set(product_prod(B,A))),Xa)),B2))),A4) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_ml(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Xa),A4,B2) ) ) ).

% fold_union_pair
tff(fact_4734_UN__finite2__subset,axiom,
    ! [A: $tType,A4: fun(nat,set(A)),B2: fun(nat,set(A)),K: nat] :
      ( ! [N2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K)))))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),A4),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),B2),top_top(set(nat))))) ) ).

% UN_finite2_subset
tff(fact_4735_cclfp__def,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,A)] : order_532582986084564980_cclfp(A,F2) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_mm(fun(A,A),fun(nat,A),F2)),top_top(set(nat)))) ) ).

% cclfp_def
tff(fact_4736_UNION__fun__upd,axiom,
    ! [B: $tType,A: $tType,A4: fun(B,set(A)),I: B,B2: set(A),J4: set(B)] :
      aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),fun_upd(B,set(A),A4,I,B2)),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)),image2(B,set(A),A4),aa(set(B),set(B),minus_minus(set(B),J4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),I),bot_bot(set(B))))))),
        $ite(member(B,I,J4),B2,bot_bot(set(A)))) ).

% UNION_fun_upd
tff(fact_4737_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),B2: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),S)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
               => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = finite_fold(A,B,F2,finite_fold(A,B,F2,Z,A4),B2) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_4738_card__UNIV__unit,axiom,
    aa(set(product_unit),nat,finite_card(product_unit),top_top(set(product_unit))) = one_one(nat) ).

% card_UNIV_unit
tff(fact_4739_top__empty__eq,axiom,
    ! [A: $tType,X2: A] :
      ( aa(A,$o,top_top(fun(A,$o)),X2)
    <=> member(A,X2,top_top(set(A))) ) ).

% top_empty_eq
tff(fact_4740_top__set__def,axiom,
    ! [A: $tType] : top_top(set(A)) = aa(fun(A,$o),set(A),collect(A),top_top(fun(A,$o))) ).

% top_set_def
tff(fact_4741_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X2: A,Xa2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X2),Xa2)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X2),Xa2),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_4742_comp__fun__commute__on_Ofun__left__comm,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Ya: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( member(A,Xa,S)
       => ( member(A,Ya,S)
         => ( aa(B,B,aa(A,fun(B,B),F2,Ya),aa(B,B,aa(A,fun(B,B),F2,Xa),Z)) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(B,B,aa(A,fun(B,B),F2,Ya),Z)) ) ) ) ) ).

% comp_fun_commute_on.fun_left_comm
tff(fact_4743_finite__update__induct,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C3: B,P: fun(fun(A,B),$o)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_mn(fun(A,B),fun(B,fun(A,$o)),F2),C3)))
     => ( aa(fun(A,B),$o,P,aTP_Lamp_ke(B,fun(A,B),C3))
       => ( ! [A5: A,B5: B,F5: fun(A,B)] :
              ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_mo(B,fun(fun(A,B),fun(A,$o)),C3),F5)))
             => ( ( aa(A,B,F5,A5) = C3 )
               => ( ( B5 != C3 )
                 => ( aa(fun(A,B),$o,P,F5)
                   => aa(fun(A,B),$o,P,fun_upd(A,B,F5,A5,B5)) ) ) ) )
         => aa(fun(A,B),$o,P,F2) ) ) ) ).

% finite_update_induct
tff(fact_4744_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => finite4664212375090638736ute_on(A,B,S,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_mp(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F2),G)) ) ).

% comp_fun_commute_on.comp_fun_commute_on_funpow
tff(fact_4745_Inter__UNIV,axiom,
    ! [A: $tType] : aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),top_top(set(set(A)))) = bot_bot(set(A)) ).

% Inter_UNIV
tff(fact_4746_SUP__UNIV__bool__expand,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: fun($o,A)] : aa(set(A),A,complete_Sup_Sup(A),aa(set($o),set(A),image2($o,A,A4),top_top(set($o)))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa($o,A,A4,$true)),aa($o,A,A4,$false)) ) ).

% SUP_UNIV_bool_expand
tff(fact_4747_Un__eq__UN,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image2($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_mq(set(A),fun(set(A),fun($o,set(A))),A4),B2)),top_top(set($o)))) ).

% Un_eq_UN
tff(fact_4748_UN__bool__eq,axiom,
    ! [A: $tType,A4: fun($o,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set($o),set(set(A)),image2($o,set(A),A4),top_top(set($o)))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa($o,set(A),A4,$true)),aa($o,set(A),A4,$false)) ).

% UN_bool_eq
tff(fact_4749_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),A4: set(A),S3: B,Ta: B,B2: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( finite4664212375090638736ute_on(A,B,S,G)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ! [X3: A] :
                  ( member(A,X3,A4)
                 => ( aa(A,fun(B,B),F2,X3) = aa(A,fun(B,B),G,X3) ) )
             => ( ( S3 = Ta )
               => ( ( A4 = B2 )
                 => ( finite_fold(A,B,F2,S3,A4) = finite_fold(A,B,G,Ta,B2) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_4750_fun__upd__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xa: B,Ya: A,A4: set(B)] :
      aa(set(B),set(A),image2(B,A,fun_upd(B,A,F2,Xa,Ya)),A4) = $ite(member(B,Xa,A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),aa(set(B),set(A),image2(B,A,F2),aa(set(B),set(B),minus_minus(set(B),A4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B)))))),aa(set(B),set(A),image2(B,A,F2),A4)) ).

% fun_upd_image
tff(fact_4751_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z,A4)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_4752_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xa),Z),A4) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_4753_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z,A4)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xa),Z),A4) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_4754_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_4755_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),Xa: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( finite_fold(A,B,F2,Z,A4) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_4756_root__def,axiom,
    ! [Na: nat,Xa: real] :
      aa(real,real,root(Na),Xa) = $ite(Na = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_mr(nat,fun(real,real),Na),Xa)) ).

% root_def
tff(fact_4757_Id__on__fold,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( id_on(A,A4) = finite_fold(A,set(product_prod(A,A)),aTP_Lamp_ms(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),bot_bot(set(product_prod(A,A))),A4) ) ) ).

% Id_on_fold
tff(fact_4758_Id__on__def,axiom,
    ! [A: $tType,A4: set(A)] : id_on(A,A4) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(A),set(set(product_prod(A,A))),image2(A,set(product_prod(A,A)),aTP_Lamp_mt(A,set(product_prod(A,A)))),A4)) ).

% Id_on_def
tff(fact_4759_top1I,axiom,
    ! [A: $tType,Xa: A] : aa(A,$o,top_top(fun(A,$o)),Xa) ).

% top1I
tff(fact_4760_top2I,axiom,
    ! [A: $tType,B: $tType,Xa: A,Ya: B] : aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),Xa),Ya) ).

% top2I
tff(fact_4761_Id__onI,axiom,
    ! [A: $tType,A3: A,A4: set(A)] :
      ( member(A,A3,A4)
     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),A3),id_on(A,A4)) ) ).

% Id_onI
tff(fact_4762_Id__on__empty,axiom,
    ! [A: $tType] : id_on(A,bot_bot(set(A))) = bot_bot(set(product_prod(A,A))) ).

% Id_on_empty
tff(fact_4763_UNIV__bool,axiom,
    top_top(set($o)) = aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$false),aa(set($o),set($o),aa($o,fun(set($o),set($o)),insert2($o),$true),bot_bot(set($o)))) ).

% UNIV_bool
tff(fact_4764_Id__on__iff,axiom,
    ! [A: $tType,Xa: A,Ya: A,A4: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),id_on(A,A4))
    <=> ( ( Xa = Ya )
        & member(A,Xa,A4) ) ) ).

% Id_on_iff
tff(fact_4765_Id__on__eqI,axiom,
    ! [A: $tType,A3: A,B3: A,A4: set(A)] :
      ( ( A3 = B3 )
     => ( member(A,A3,A4)
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),id_on(A,A4)) ) ) ).

% Id_on_eqI
tff(fact_4766_Id__onE,axiom,
    ! [A: $tType,C3: product_prod(A,A),A4: set(A)] :
      ( member(product_prod(A,A),C3,id_on(A,A4))
     => ~ ! [X3: A] :
            ( member(A,X3,A4)
           => ( C3 != aa(A,product_prod(A,A),product_Pair(A,A,X3),X3) ) ) ) ).

% Id_onE
tff(fact_4767_Id__on__def_H,axiom,
    ! [A: $tType,A4: fun(A,$o)] : id_on(A,aa(fun(A,$o),set(A),collect(A),A4)) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_mu(fun(A,$o),fun(A,fun(A,$o)),A4))) ).

% Id_on_def'
tff(fact_4768_DERIV__real__root__generic,axiom,
    ! [Na: nat,Xa: real,D3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( Xa != zero_zero(real) )
       => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
             => ( D3 = 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),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
               => ( D3 = 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),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
               => ( D3 = 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),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(Na),D3,topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_4769_DERIV__even__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
         => has_field_derivative(real,root(Na),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),Na))),aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_4770_DERIV__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_mv(real,real),suminf(real,aTP_Lamp_mw(real,fun(nat,real),Xa)),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_4771_at__within__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A] : topolo174197925503356063within(A,A3,bot_bot(set(A))) = bot_bot(filter(A)) ) ).

% at_within_empty
tff(fact_4772_deriv__nonneg__imp__mono,axiom,
    ! [A3: real,B3: real,G: fun(real,real),G4: fun(real,real)] :
      ( ! [X3: real] :
          ( member(real,X3,set_or1337092689740270186AtMost(real,A3,B3))
         => has_field_derivative(real,G,aa(real,real,G4,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ( ! [X3: real] :
            ( member(real,X3,set_or1337092689740270186AtMost(real,A3,B3))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,G4,X3)) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,G,A3)),aa(real,real,G,B3)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_4773_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y2) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,A3)),aa(real,real,F2,B3)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_4774_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,B3)),aa(real,real,F2,A3)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_4775_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F9: A,Xa: A,S3: set(A),Ta: set(A)] :
          ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,Xa,S3))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S3)
           => has_field_derivative(A,F2,F9,topolo174197925503356063within(A,Xa,Ta)) ) ) ) ).

% DERIV_subset
tff(fact_4776_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Ya: A,Xa: A,S3: set(A),Ta: set(A)] :
          ( has_field_derivative(A,F2,Ya,topolo174197925503356063within(A,Xa,S3))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S3)
           => has_field_derivative(A,F2,Ya,topolo174197925503356063within(A,Xa,Ta)) ) ) ) ).

% has_field_derivative_subset
tff(fact_4777_at__within__union,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A,S: set(A),T3: set(A)] : topolo174197925503356063within(A,Xa,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),topolo174197925503356063within(A,Xa,S)),topolo174197925503356063within(A,Xa,T3)) ) ).

% at_within_union
tff(fact_4778_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F3: filter(A)] : has_field_derivative(A,aTP_Lamp_mx(A,fun(A,A),K),zero_zero(A),F3) ) ).

% DERIV_const
tff(fact_4779_has__real__derivative__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,minus_minus(real,Xa),H3),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D6)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,minus_minus(real,Xa),H3))),aa(real,real,F2,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_4780_has__real__derivative__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,minus_minus(real,Xa),H3),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D6)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xa)),aa(real,real,F2,aa(real,real,minus_minus(real,Xa),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_4781_has__real__derivative__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D6)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xa)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_4782_has__real__derivative__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,S: set(real)] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,S))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3),S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D6)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3))),aa(real,real,F2,Xa)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_4783_DERIV__neg__imp__decreasing,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B3)),aa(real,real,F2,A3)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_4784_DERIV__pos__imp__increasing,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A3)),aa(real,real,F2,B3)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_4785_DERIV__pos__inc__right,axiom,
    ! [F2: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D6)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xa)),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_4786_DERIV__neg__dec__right,axiom,
    ! [F2: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D6)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xa),H3))),aa(real,real,F2,Xa)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_4787_DERIV__pos__inc__left,axiom,
    ! [F2: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D6)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,aa(real,real,minus_minus(real,Xa),H3))),aa(real,real,F2,Xa)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_4788_DERIV__neg__dec__left,axiom,
    ! [F2: fun(real,real),L: real,Xa: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [D6: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D6)
            & ! [H3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H3),D6)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,Xa)),aa(real,real,F2,aa(real,real,minus_minus(real,Xa),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_4789_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xa: A,S3: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xa,S3))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,Xa,S3))
           => ( ( aa(A,A,G,Xa) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_my(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(A,A,G,Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F2,Xa)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,Xa)),aa(A,A,G,Xa))),topolo174197925503356063within(A,Xa,S3)) ) ) ) ) ).

% DERIV_divide
tff(fact_4790_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xa: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xa,S3))
         => ( ( aa(A,A,F2,Xa) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_mz(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,Xa))),D3)),aa(A,A,inverse_inverse(A),aa(A,A,F2,Xa)))),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% DERIV_inverse'
tff(fact_4791_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),Ta: set(A),Xa: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),Ta)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),topolo174197925503356063within(A,Xa,S3)),topolo174197925503356063within(A,Xa,Ta)) ) ) ).

% at_le
tff(fact_4792_MVT2,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),F9: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
             => has_field_derivative(real,F2,aa(real,real,F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B3)
            & ( aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B3),A3)),aa(real,real,F9,Z2)) ) ) ) ) ).

% MVT2
tff(fact_4793_DERIV__local__const,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xa),Y))),D2)
             => ( aa(real,real,F2,Xa) = aa(real,real,F2,Y) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_4794_DERIV__ln,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xa),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_4795_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xa: A,S3: set(A)] :
          ( ( Xa != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xa,S3)) ) ) ).

% DERIV_inverse
tff(fact_4796_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xa: A,S3: set(A),Na: nat] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xa,S3))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_na(fun(A,A),fun(nat,fun(A,A)),F2),Na),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Na)),aa(A,A,aa(A,fun(A,A),times_times(A),D3),aa(nat,A,power_power(A,aa(A,A,F2,Xa)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xa,S3)) ) ) ).

% DERIV_power
tff(fact_4797_DERIV__local__min,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xa),Y))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Xa)),aa(real,real,F2,Y)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_4798_DERIV__local__max,axiom,
    ! [F2: fun(real,real),L: real,Xa: real,D2: real] :
      ( has_field_derivative(real,F2,L,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
       => ( ! [Y: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xa),Y))),D2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,Y)),aa(real,real,F2,Xa)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_4799_DERIV__ln__divide,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,ln_ln(real),divide_divide(real,one_one(real),Xa),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_4800_DERIV__pow,axiom,
    ! [Na: nat,Xa: real,S3: set(real)] : has_field_derivative(real,aTP_Lamp_nb(nat,fun(real,real),Na),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Na)),aa(nat,real,power_power(real,Xa),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xa,S3)) ).

% DERIV_pow
tff(fact_4801_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,Xa: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B3)
           => ( topolo174197925503356063within(A,Xa,set_or1337092689740270186AtMost(A,A3,B3)) = topolo174197925503356063within(A,Xa,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_4802_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( topolo174197925503356063within(A,B3,set_or1337092689740270186AtMost(A,A3,B3)) = topolo174197925503356063within(A,B3,aa(A,set(A),set_ord_lessThan(A),B3)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_4803_trivial__limit__at__left__bot,axiom,
    ! [A: $tType] :
      ( ( order_bot(A)
        & topolo1944317154257567458pology(A) )
     => ( topolo174197925503356063within(A,bot_bot(A),aa(A,set(A),set_ord_lessThan(A),bot_bot(A))) = bot_bot(filter(A)) ) ) ).

% trivial_limit_at_left_bot
tff(fact_4804_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xa: A,S3: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xa,S3))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,Xa,S3))
           => ( ( aa(A,A,G,Xa) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_my(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,G,Xa))),aa(A,A,aa(A,fun(A,A),times_times(A),E2),aa(A,A,F2,Xa))),aa(nat,A,power_power(A,aa(A,A,G,Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xa,S3)) ) ) ) ) ).

% DERIV_quotient
tff(fact_4805_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xa: A,S3: set(A)] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xa,S3))
         => ( ( aa(A,A,F2,Xa) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_mz(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,power_power(A,aa(A,A,F2,Xa)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_4806_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K4: real,C3: fun(nat,A),F2: fun(A,A),F9: A,Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K4)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),Z2),aa(A,A,F2,Z2)) )
         => ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K4)
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_nc(fun(nat,A),fun(A,fun(nat,A)),C3),Z),F9) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_4807_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_nd(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,minus_minus(real,R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_4808_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K4: real,C3: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K4)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),Z2)) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K4)
           => has_field_derivative(A,aTP_Lamp_ne(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_nc(fun(nat,A),fun(A,fun(nat,A)),C3),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_4809_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K4: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K4))
           => has_field_derivative(A,aTP_Lamp_ne(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_nc(fun(nat,A),fun(A,fun(nat,A)),C3),Xa)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_4810_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K4: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_nc(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_nf(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K4))
               => has_field_derivative(A,aTP_Lamp_ne(fun(nat,A),fun(A,A),C3),suminf(A,aa(A,fun(nat,A),aTP_Lamp_nc(fun(nat,A),fun(A,fun(nat,A)),C3),Xa)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_4811_DERIV__log,axiom,
    ! [Xa: real,B3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,log(B3),divide_divide(real,one_one(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B3)),Xa)),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_4812_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,Xa: real,R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xa))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_ng(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,Xa),aa(real,real,minus_minus(real,R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_4813_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,Xa: real,F2: fun(real,real),R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xa,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,G,Xa))
       => ( has_field_derivative(real,F2,R2,topolo174197925503356063within(real,Xa,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_nh(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,Xa),aa(real,real,F2,Xa))),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,Xa)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F2,Xa)),aa(real,real,G,Xa)))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_4814_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,cos(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_4815_DERIV__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
     => has_field_derivative(real,sqrt,divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa)),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_4816_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sin(A,Xa) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,sin(A,Xa)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_4817_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Xa: A,Db: A,S3: set(A)] :
          ( ( cosh(A,aa(A,A,G,Xa)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xa,S3))
           => has_field_derivative(A,aTP_Lamp_ni(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,aa(A,A,tanh(A),aa(A,A,G,Xa))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_4818_DERIV__real__sqrt__generic,axiom,
    ! [Xa: real,D3: real] :
      ( ( Xa != zero_zero(real) )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
         => ( D3 = divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa)),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),zero_zero(real))
           => ( D3 = divide_divide(real,aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xa))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D3,topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_4819_arcosh__real__has__field__derivative,axiom,
    ! [Xa: real,A4: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => has_field_derivative(real,arcosh(real),divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xa,A4)) ) ).

% arcosh_real_has_field_derivative
tff(fact_4820_artanh__real__has__field__derivative,axiom,
    ! [Xa: real,A4: set(real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => has_field_derivative(real,artanh(real),divide_divide(real,one_one(real),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xa,A4)) ) ).

% artanh_real_has_field_derivative
tff(fact_4821_DERIV__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Xa)
       => has_field_derivative(real,root(Na),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),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_4822_DERIV__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),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,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_4823_DERIV__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_4824_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xa: real,Na: 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)))) )
     => ? [T5: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa))
          & ( aa(real,real,F2,Xa) = 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_nj(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_4825_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Xa: real,Na: 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))))
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa))
            & ( aa(real,real,F2,Xa) = 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_nj(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_4826_DERIV__odd__real__root,axiom,
    ! [Na: nat,Xa: real] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
     => ( ( Xa != zero_zero(real) )
       => has_field_derivative(real,root(Na),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),Na)),aa(nat,real,power_power(real,aa(real,real,root(Na),Xa)),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_4827_Maclaurin__minus,axiom,
    ! [H: real,Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),zero_zero(real))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M4: nat,T5: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),H),T5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),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)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
           => ? [T5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),H),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),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_nk(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_4828_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T5: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),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)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ? [T5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),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_nk(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_4829_Maclaurin,axiom,
    ! [H: real,Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),H)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
         => ( ! [M4: nat,T5: real] :
                ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),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)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
           => ? [T5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),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_nk(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,H),Na))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_4830_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat,Xa: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( ( Xa != 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))))
           => ? [T5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T5))
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa))
                & ( aa(real,real,F2,Xa) = 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_nj(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_4831_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F2: fun(real,real),Na: nat,Xa: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
     => ( ! [M4: nat,T5: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa)) )
           => 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)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
       => ? [T5: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T5)),aa(real,real,abs_abs(real),Xa))
            & ( aa(real,real,F2,Xa) = 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_nj(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,Xa),Na))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_4832_Taylor__down,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B3: real,C3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T5: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),B3) )
             => 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)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),B3)
             => ? [T5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),T5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),C3)
                  & ( aa(real,real,F2,A3) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_nl(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A3),C3)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,A3),C3)),Na))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_4833_Taylor__up,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B3: real,C3: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T5: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),B3) )
             => 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)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),B3)
             => ? [T5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),T5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),B3)
                  & ( aa(real,real,F2,B3) = 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_nl(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B3),C3)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,B3),C3)),Na))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_4834_Taylor,axiom,
    ! [Na: nat,Diff: fun(nat,fun(real,real)),F2: fun(real,real),A3: real,B3: real,C3: real,Xa: real] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F2 )
       => ( ! [M4: nat,T5: real] :
              ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),T5)
                & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),B3) )
             => 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)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),C3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C3),B3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Xa)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),B3)
                 => ( ( Xa != C3 )
                   => ? [T5: real] :
                        ( $ite(
                            aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),C3),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),T5)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),C3) ),
                            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),T5)
                            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),T5),Xa) ) )
                        & ( aa(real,real,F2,Xa) = 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_nm(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C3),Xa)),aa(nat,set(nat),set_ord_lessThan(nat),Na))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,Na),T5),semiring_char_0_fact(real,Na))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Xa),C3)),Na))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_4835_Maclaurin__lemma2,axiom,
    ! [Na: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B2: real] :
      ( ! [M4: nat,T5: real] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T5)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T5),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)),T5),topolo174197925503356063within(real,T5,top_top(set(real)))) )
     => ( ( Na = aa(nat,nat,suc,K) )
       => ! [M3: nat,T7: real] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),Na)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T7)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T7),H) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_no(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Na),Diff),B2),M3),aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T7)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_np(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M3),T7)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M3))))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),divide_divide(real,aa(nat,real,power_power(real,T7),aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M3))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Na),aa(nat,nat,suc,M3))))))),topolo174197925503356063within(real,T7,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_4836_DERIV__power__series_H,axiom,
    ! [R: real,F2: fun(nat,real),X0: real] :
      ( ! [X3: real] :
          ( member(real,X3,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_nq(fun(nat,real),fun(real,fun(nat,real)),F2),X3)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => has_field_derivative(real,aTP_Lamp_ns(fun(nat,real),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),aTP_Lamp_nq(fun(nat,real),fun(real,fun(nat,real)),F2),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_4837_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),S3: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xa)),one_one(real))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,S3))
             => has_derivative(A,real,aTP_Lamp_nt(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_nu(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,S3)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_4838_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),S3: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xa))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,G,Xa)),one_one(real))
           => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,S3))
             => has_derivative(A,real,aTP_Lamp_nv(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_nw(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,S3)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_4839_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_4840_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_4841_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B3) = bot_bot(set(A)) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_4842_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B3) )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_4843_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% infinite_Ioo_iff
tff(fact_4844_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Xa,Ya)) = Ya ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_4845_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Ya,Xa)) = Xa ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_4846_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Xa,Ya)) = Xa ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_4847_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Ya,Xa)) = Ya ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_4848_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C3: B,F3: filter(A)] : has_derivative(A,B,aTP_Lamp_nx(B,fun(A,B),C3),aTP_Lamp_ny(A,B),F3) ) ).

% has_derivative_const
tff(fact_4849_has__derivative__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xa: A,S3: set(A),Ta: set(A)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S3))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S3)
           => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,Ta)) ) ) ) ).

% has_derivative_subset
tff(fact_4850_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(set(A),$o,finite_finite2(A),set_or5935395276787703475ssThan(A,A3,B3)) ) ) ).

% infinite_Ioo
tff(fact_4851_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,B),Xa: A] :
          ( has_derivative(A,B,aTP_Lamp_ny(A,B),F3,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => ! [X2: A] : aa(A,B,F3,X2) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_4852_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Ta: set(A),G: fun(A,B),G4: fun(A,fun(A,B)),F2: fun(C,A),S3: set(C),Xa: C,F9: fun(C,A)] :
          ( ! [X3: A] :
              ( member(A,X3,Ta)
             => has_derivative(A,B,G,aa(A,fun(A,B),G4,X3),topolo174197925503356063within(A,X3,Ta)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),S3)),Ta)
           => ( member(C,Xa,S3)
             => ( has_derivative(C,A,F2,F9,topolo174197925503356063within(C,Xa,S3))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_nz(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_oa(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G4),F2),Xa),F9),topolo174197925503356063within(C,Xa,S3)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_4853_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),set_or5935395276787703475ssThan(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_4854_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(4)
tff(fact_4855_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(5)
tff(fact_4856_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(1)
tff(fact_4857_ivl__disj__int__one_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(1)
tff(fact_4858_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4859_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),set_or7035219750837199246ssThan(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_4860_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_4861_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A3,B3) ) ).

% atLeastAtMost_diff_ends
tff(fact_4862_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_4863_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xa: A,S: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xa,S))
           => ( ( aa(A,B,G,Xa) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ob(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_oc(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F9),Xa),G),G4),topolo174197925503356063within(A,Xa,S)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_4864_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [F2: fun(B,A),Xa: B,F9: fun(B,A),S: set(B)] :
          ( ( aa(B,A,F2,Xa) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F9,topolo174197925503356063within(B,Xa,S))
           => has_derivative(B,A,aTP_Lamp_od(fun(B,A),fun(B,A),F2),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_oe(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),F2),Xa),F9),topolo174197925503356063within(B,Xa,S)) ) ) ) ).

% has_derivative_inverse
tff(fact_4865_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A,S: set(A)] :
          ( ( Xa != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_of(A,fun(A,A),Xa),topolo174197925503356063within(A,Xa,S)) ) ) ).

% has_derivative_inverse'
tff(fact_4866_DERIV__isconst3,axiom,
    ! [A3: real,B3: real,Xa: real,Ya: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( member(real,Xa,set_or5935395276787703475ssThan(real,A3,B3))
       => ( member(real,Ya,set_or5935395276787703475ssThan(real,A3,B3))
         => ( ! [X3: real] :
                ( member(real,X3,set_or5935395276787703475ssThan(real,A3,B3))
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) )
           => ( aa(real,real,F2,Xa) = aa(real,real,F2,Ya) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_4867_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_4868_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_4869_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),S3: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xa))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,S3))
           => has_derivative(A,real,aTP_Lamp_og(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_oh(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% has_derivative_ln
tff(fact_4870_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xa: A,S: set(A),G: fun(A,B),G4: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S))
         => ( has_derivative(A,B,G,G4,topolo174197925503356063within(A,Xa,S))
           => ( ( aa(A,B,G,Xa) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oi(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_oj(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),F2),F9),Xa),G),G4),topolo174197925503356063within(A,Xa,S)) ) ) ) ) ).

% has_derivative_divide
tff(fact_4871_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I5: set(A),F2: fun(A,fun(B,C)),F9: fun(A,fun(B,C)),Xa: B,S: set(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => has_derivative(B,C,aa(A,fun(B,C),F2,I2),aa(A,fun(B,C),F9,I2),topolo174197925503356063within(B,Xa,S)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ol(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F2),aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_on(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),F2),F9),Xa),topolo174197925503356063within(B,Xa,S)) ) ) ).

% has_derivative_prod
tff(fact_4872_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G4: fun(A,real),Xa: A,X4: set(A),F2: fun(A,real),F9: fun(A,real)] :
          ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,X4))
         => ( has_derivative(A,real,F2,F9,topolo174197925503356063within(A,Xa,X4))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xa))
             => ( member(A,Xa,X4)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_oo(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_op(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G4),Xa),F2),F9),topolo174197925503356063within(A,Xa,X4)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_4873_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xa: A,G4: fun(A,real),S3: set(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,Xa))
         => ( has_derivative(A,real,G,G4,topolo174197925503356063within(A,Xa,S3))
           => has_derivative(A,real,aTP_Lamp_oq(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_or(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xa),G4),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_4874_DERIV__series_H,axiom,
    ! [F2: fun(real,fun(nat,real)),F9: fun(real,fun(nat,real)),X0: real,A3: real,B3: real,L4: fun(nat,real)] :
      ( ! [N2: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_os(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F2),N2),aa(nat,real,aa(real,fun(nat,real),F9,X0),N2),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X3: real] :
            ( member(real,X3,set_or5935395276787703475ssThan(real,A3,B3))
           => summable(real,aa(real,fun(nat,real),F2,X3)) )
       => ( member(real,X0,set_or5935395276787703475ssThan(real,A3,B3))
         => ( summable(real,aa(real,fun(nat,real),F9,X0))
           => ( summable(real,L4)
             => ( ! [N2: nat,X3: real,Y: real] :
                    ( member(real,X3,set_or5935395276787703475ssThan(real,A3,B3))
                   => ( member(real,Y,set_or5935395276787703475ssThan(real,A3,B3))
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(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,L4,N2)),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X3),Y)))) ) )
               => has_field_derivative(real,aTP_Lamp_ot(fun(real,fun(nat,real)),fun(real,real),F2),suminf(real,aa(real,fun(nat,real),F9,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_4875_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K4: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_nf(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K4))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ov(fun(nat,A),fun(A,fun(A,A)),C3),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_4876_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C3: fun(nat,A),K4: A,Xa: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),C3),K4))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),real_V7770717601297561774m_norm(A,K4))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),aTP_Lamp_ne(fun(nat,A),fun(A,A),C3)) ) ) ) ).

% isCont_powser
tff(fact_4877_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A3: A,F2: fun(A,B),C3: fun(nat,B),K4: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_ow(fun(nat,B),fun(B,fun(nat,B)),C3),K4))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,A3))),real_V7770717601297561774m_norm(B,K4))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_oy(fun(A,B),fun(fun(nat,B),fun(A,B)),F2),C3)) ) ) ) ) ).

% isCont_powser'
tff(fact_4878_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or5935395276787703475ssThan(nat,L,U)) ).

% finite_greaterThanLessThan
tff(fact_4879_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_oz(A,fun(fun(B,A),fun(B,A)),C3),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),L)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_4880_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F2: fun(B,A),L: A,F3: filter(B)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_pa(A,fun(fun(B,A),fun(B,A)),C3),F2),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C3)),F3)
          <=> filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,L),F3) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_4881_power__tendsto__0__iff,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pb(nat,fun(fun(A,real),fun(A,real)),Na),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_4882_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xa: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xa,top_top(set(A))),F2)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pc(A,fun(fun(A,B),fun(A,B)),Xa),F2),topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xa)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_4883_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,A3: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_pd(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A3,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_4884_filterlim__sup,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(B),F1: filter(A),F22: filter(A)] :
      ( filterlim(A,B,F2,F3,F1)
     => ( filterlim(A,B,F2,F3,F22)
       => filterlim(A,B,F2,F3,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F1),F22)) ) ) ).

% filterlim_sup
tff(fact_4885_tendsto__arcosh,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),A3)
       => filterlim(A,real,aTP_Lamp_pe(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F3) ) ) ).

% tendsto_arcosh
tff(fact_4886_filterlim__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(A)] : filterlim(A,B,F2,top_top(filter(B)),F3) ).

% filterlim_top
tff(fact_4887_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F22: filter(B),F1: filter(A),F23: filter(B),F12: filter(A)] :
      ( filterlim(A,B,F2,F22,F1)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F22),F23)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F12),F1)
         => filterlim(A,B,F2,F23,F12) ) ) ) ).

% filterlim_mono
tff(fact_4888_tendsto__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F3: filter(A),F8: filter(A),F2: fun(A,B),L: B] :
          ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F8)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F8)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% tendsto_mono
tff(fact_4889_filterlim__inf,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F22: filter(B),F32: filter(B),F1: filter(A)] :
      ( filterlim(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F22),F32),F1)
    <=> ( filterlim(A,B,F2,F22,F1)
        & filterlim(A,B,F2,F32,F1) ) ) ).

% filterlim_inf
tff(fact_4890_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A3: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A3),F3)
         => ( ( cos(A,A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_pf(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A3)),F3) ) ) ) ).

% tendsto_tan
tff(fact_4891_filterlim__ident,axiom,
    ! [A: $tType,F3: filter(A)] : filterlim(A,A,aTP_Lamp_jq(A,A),F3,F3) ).

% filterlim_ident
tff(fact_4892_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F32: filter(B),F22: filter(A),F2: fun(C,A),F1: filter(C)] :
      ( filterlim(A,B,G,F32,F22)
     => ( filterlim(C,A,F2,F22,F1)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pg(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2),F32,F1) ) ) ).

% filterlim_compose
tff(fact_4893_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_ph(fun(A,B),fun(A,real),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_norm_zero
tff(fact_4894_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_ph(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_4895_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_ph(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_4896_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),F3: filter(A),C3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_pi(fun(A,B),fun(B,fun(A,B)),F2),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_divide_zero
tff(fact_4897_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F3)
           => ( ( B3 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pj(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,divide_divide(B,A3,B3)),F3) ) ) ) ) ).

% tendsto_divide
tff(fact_4898_tendsto__sgn,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)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_pk(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,sgn_sgn(B,L)),F3) ) ) ) ).

% tendsto_sgn
tff(fact_4899_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( ( A3 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_pl(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A3)),F3) ) ) ) ).

% tendsto_inverse
tff(fact_4900_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pm(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_mult_zero
tff(fact_4901_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_pn(fun(A,B),fun(B,fun(A,B)),F2),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_left_zero
tff(fact_4902_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),C3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_po(fun(A,B),fun(B,fun(A,B)),F2),C3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% tendsto_mult_right_zero
tff(fact_4903_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pp(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_add_zero
tff(fact_4904_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),F2: fun(B,fun(A,C)),F3: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_pq(fun(B,fun(A,C)),fun(A,fun(B,C)),F2),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_pr(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),F2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ).

% tendsto_null_sum
tff(fact_4905_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),G: fun(A,B),F3: filter(A),A3: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ps(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F3) ) ) ) ).

% Lim_transform_eq
tff(fact_4906_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_pt(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_4907_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ps(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F3) ) ) ) ).

% Lim_transform2
tff(fact_4908_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A3: B,F3: filter(A),F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pu(fun(A,B),fun(fun(A,B),fun(A,B)),G),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3) ) ) ) ).

% Lim_transform
tff(fact_4909_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_pt(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_4910_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_pt(fun(A,B),fun(B,fun(A,B)),F2),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ).

% LIM_zero
tff(fact_4911_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( ( cosh(B,A3) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_pv(fun(A,B),fun(A,B),F2),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A3)),F3) ) ) ) ).

% tendsto_tanh
tff(fact_4912_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(A,A),A3: A,F3: filter(A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,A3),F3)
         => ( ( sin(A,A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_pw(fun(A,A),fun(A,A),F2),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A3)),F3) ) ) ) ).

% tendsto_cot
tff(fact_4913_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,real),A3: A,G: fun(A,real)] :
          ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A3 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,G,X3)) )
           => ( ! [X3: A] :
                  ( ( X3 != A3 )
                 => aa(real,$o,aa(real,fun(real,$o),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,A3,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_4914_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A3),top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X3: A] :
                      ( ( ( X3 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A3))),D4) )
                     => ( aa(A,B,F2,X3) != aa(A,B,F2,A3) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_px(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_4915_tendsto__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(A,B),L: filter(B),Xa: A,S: set(A),T3: set(A)] :
          ( filterlim(A,B,F2,L,topolo174197925503356063within(A,Xa,S))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S)
           => filterlim(A,B,F2,L,topolo174197925503356063within(A,Xa,T3)) ) ) ) ).

% tendsto_within_subset
tff(fact_4916_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_py(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_4917_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L4: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_py(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_4918_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A,L4: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_py(fun(A,B),fun(A,fun(A,B)),F2),A3),topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_4919_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,B),F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_pz(fun(A,B),fun(nat,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_null_power
tff(fact_4920_tendsto__log,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
         => ( ( A3 != one_one(real) )
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qa(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A3),B3)),F3) ) ) ) ) ) ).

% tendsto_log
tff(fact_4921_tendsto__artanh,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),one_one(real))
         => filterlim(A,real,aTP_Lamp_qb(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A3)),F3) ) ) ) ).

% tendsto_artanh
tff(fact_4922_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L: B,A3: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A3 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(C,C,minus_minus(C,aa(A,C,G,X3)),M))),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X3)),L))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_4923_LIM__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A3: B,F2: fun(B,C),L4: C] :
          ( nO_MATCH(A,B,zero_zero(A),A3)
         => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L4),topolo174197925503356063within(B,A3,top_top(set(B))))
          <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_qc(B,fun(fun(B,C),fun(B,C)),A3),F2),topolo7230453075368039082e_nhds(C,L4),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_4924_filterlim__INF_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,Xa: A,A4: set(A),F2: fun(B,C),F3: filter(C),G2: fun(A,filter(B))] :
      ( member(A,Xa,A4)
     => ( filterlim(B,C,F2,F3,aa(A,filter(B),G2,Xa))
       => filterlim(B,C,F2,F3,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),G2),A4))) ) ) ).

% filterlim_INF'
tff(fact_4925_filterlim__INF,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G2: fun(C,filter(B)),B2: set(C),F3: filter(A)] :
      ( filterlim(A,B,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),G2),B2)),F3)
    <=> ! [X: C] :
          ( member(C,X,B2)
         => filterlim(A,B,F2,aa(C,filter(B),G2,X),F3) ) ) ).

% filterlim_INF
tff(fact_4926_IVT,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),A3: B,Ya: A,B3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,A3)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),aa(B,A,F2,B3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B3)
             => ( ! [X3: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X3)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B3) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X3,top_top(set(B))),F2) )
               => ? [X3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X3)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B3)
                    & ( aa(B,A,F2,X3) = Ya ) ) ) ) ) ) ) ).

% IVT
tff(fact_4927_IVT2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),B3: B,Ya: A,A3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,B3)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),aa(B,A,F2,A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B3)
             => ( ! [X3: B] :
                    ( ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X3)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B3) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X3,top_top(set(B))),F2) )
               => ? [X3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X3)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B3)
                    & ( aa(B,A,F2,X3) = Ya ) ) ) ) ) ) ) ).

% IVT2
tff(fact_4928_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L4: B,A3: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [S4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S4)
                & ! [X2: A] :
                    ( ( ( X2 != A3 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A3))),S4) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X2)),L4))),R2) ) ) ) ) ) ).

% LIM_D
tff(fact_4929_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B),L4: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S8: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S8)
                  & ! [X3: A] :
                      ( ( ( X3 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A3))),S8) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X3)),L4))),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_4930_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),L4: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A] :
                      ( ( ( X != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),A3))),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F2,X)),L4))),R5) ) ) ) ) ) ).

% LIM_eq
tff(fact_4931_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A3: A,F2: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
         => ( ! [X3: A] :
                ( ( X3 != A3 )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A3))),R)
                 => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_4932_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(A,A),A3: A,D3: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qd(fun(A,A),fun(A,fun(A,A)),F2),A3),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qe(fun(A,A),fun(A,fun(A,A)),F2),A3),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_4933_isCont__Lb__Ub,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
     => ( ! [X3: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
       => ? [L5: real,M9: real] :
            ( ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B3) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L5),aa(real,real,F2,X2))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F2,X2)),M9) ) )
            & ! [Y2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L5),Y2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y2),M9) )
               => ? [X3: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
                    & ( aa(real,real,F2,X3) = Y2 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_4934_LIM__fun__gt__zero,axiom,
    ! [F2: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),L)
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X2: real] :
                ( ( ( X2 != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C3),X2))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F2,X2)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_4935_LIM__fun__not__zero,axiom,
    ! [F2: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X2: real] :
                ( ( ( X2 != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C3),X2))),R3) )
               => ( aa(real,real,F2,X2) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_4936_LIM__fun__less__zero,axiom,
    ! [F2: fun(real,real),L: real,C3: real] :
      ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C3,top_top(set(real))))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),L),zero_zero(real))
       => ? [R3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
            & ! [X2: real] :
                ( ( ( X2 != C3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C3),X2))),R3) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,X2)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_4937_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B3: B,A3: A,G: fun(B,C),C3: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B3),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(B,B3,top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X3: A] :
                      ( ( ( X3 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A3))),D4) )
                     => ( aa(A,B,F2,X3) != B3 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_px(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_4938_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),G)
           => ( ( aa(A,B,G,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),aa(fun(A,B),fun(A,B),aTP_Lamp_qf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_4939_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A3: A,S3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),aTP_Lamp_qg(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_4940_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,S3: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),aTP_Lamp_qh(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_4941_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xa: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qi(fun(A,A),fun(A,fun(A,A)),F2),Xa),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_4942_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xa: A] :
          ( has_field_derivative(A,F2,D3,topolo174197925503356063within(A,Xa,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qi(fun(A,A),fun(A,fun(A,A)),F2),Xa),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_4943_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_qj(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_4944_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F2: fun(A,B),K4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( ! [H4: A] :
                ( ( H4 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H4)),K)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,H4))),aa(real,real,aa(real,fun(real,real),times_times(real),K4),real_V7770717601297561774m_norm(A,H4))) ) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_4945_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M9: A] :
                ( ! [X2: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X2)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B3) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M9),aa(real,A,F2,X2)) )
                & ? [X3: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
                    & ( aa(real,A,F2,X3) = M9 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_4946_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M9: A] :
                ( ! [X2: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X2)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B3) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X2)),M9) )
                & ? [X3: real] :
                    ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
                    & ( aa(real,A,F2,X3) = M9 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_4947_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M9: A] :
              ! [X2: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X2)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B3) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X2)),M9) ) ) ) ) ).

% isCont_bounded
tff(fact_4948_isCont__inverse__function2,axiom,
    ! [A3: real,Xa: real,B3: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B3)
       => ( ! [Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Z2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z2),B3)
               => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Z2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z2),B3)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xa),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_4949_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D3: A,Xa: A] :
          ( has_derivative(A,A,F2,aa(A,fun(A,A),times_times(A),D3),topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qi(fun(A,A),fun(A,fun(A,A)),F2),Xa),topolo7230453075368039082e_nhds(A,D3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_4950_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,F2: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( ( aa(A,B,G,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_qf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% isCont_divide
tff(fact_4951_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_qh(fun(A,B),fun(A,B),F2)) ) ) ) ).

% isCont_sgn
tff(fact_4952_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(A,B),F3: filter(B),A3: A] :
          ( filterlim(A,B,F2,F3,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qk(fun(A,B),fun(A,fun(A,B)),F2),A3),F3,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_4953_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,S3: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,S3),F2)
         => ( ( cos(A,aa(A,A,F2,Xa)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,S3),aTP_Lamp_pf(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_tan
tff(fact_4954_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,minus_minus(nat,J),aa(nat,nat,suc,I)))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J))),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_4955_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A,S3: set(A),F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,S3),F2)
         => ( ( sin(A,aa(A,A,F2,Xa)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,S3),aTP_Lamp_pw(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_within_cot
tff(fact_4956_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Xa: A,A4: set(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xa,A4),F2)
         => ( ( cosh(B,aa(A,B,F2,Xa)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xa,A4),aTP_Lamp_ql(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_4957_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,F2: fun(real,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
           => ? [M9: A] :
                ( ! [X2: real] :
                    ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X2)
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B3) )
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(real,A,F2,X2)),M9) )
                & ! [N7: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),M9)
                   => ? [X3: real] :
                        ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less(A),N7),aa(real,A,F2,X3)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_4958_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cos(A,Xa) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_4959_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( sin(A,Xa) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_4960_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( ( cosh(A,Xa) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xa,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_4961_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: real,A3: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
         => ( ! [X3: A] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S3)
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),A3),X3),aa(A,A,F2,X3)) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_4962_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: real,A3: fun(nat,A),F2: fun(A,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S3)
         => ( ! [X3: A] :
                ( ( X3 != zero_zero(A) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S3)
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),A3),X3),aa(A,A,F2,X3)) ) )
           => filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_4963_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F2: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K)
         => ( summable(real,F2)
           => ( ! [H4: A,N2: nat] :
                  ( ( H4 != zero_zero(A) )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,H4)),K)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H4),N2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F2,N2)),real_V7770717601297561774m_norm(A,H4))) ) )
             => filterlim(A,B,aTP_Lamp_qm(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_4964_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( cos(A,aa(A,A,F2,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_pf(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_tan'
tff(fact_4965_isCont__arcosh,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_4966_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S3),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S3),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A3))
             => ( ( aa(A,real,F2,A3) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,A3))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S3),aa(fun(A,real),fun(A,real),aTP_Lamp_qn(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_4967_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,F2: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( ( sin(A,aa(A,A,F2,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_pw(fun(A,A),fun(A,A),F2)) ) ) ) ).

% isCont_cot'
tff(fact_4968_DERIV__inverse__function,axiom,
    ! [F2: fun(real,real),D3: real,G: fun(real,real),Xa: real,A3: real,B3: real] :
      ( has_field_derivative(real,F2,D3,topolo174197925503356063within(real,aa(real,real,G,Xa),top_top(set(real))))
     => ( ( D3 != zero_zero(real) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xa)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),B3)
           => ( ! [Y: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Y)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y),B3)
                   => ( aa(real,real,F2,aa(real,real,G,Y)) = Y ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D3),topolo174197925503356063within(real,Xa,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_4969_isCont__arccos,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_4970_isCont__arcsin,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_4971_LIM__less__bound,axiom,
    ! [B3: real,Xa: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B3),Xa)
     => ( ! [X3: real] :
            ( member(real,X3,set_or5935395276787703475ssThan(real,B3,Xa))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,X3)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,F2,Xa)) ) ) ) ).

% LIM_less_bound
tff(fact_4972_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,A3))
             => ( ( aa(A,real,F2,A3) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,A3))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_qn(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_4973_isCont__artanh,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xa,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_4974_isCont__inverse__function,axiom,
    ! [D2: real,Xa: real,G: fun(real,real),F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
     => ( ! [Z2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z2),Xa))),D2)
           => ( aa(real,real,G,aa(real,real,F2,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z2),Xa))),D2)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F2,Xa),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_4975_GMVT_H,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),G: fun(real,real),G4: fun(real,real),F9: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ! [Z2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Z2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z2),B3)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F2) ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Z2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Z2),B3)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),G) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z2)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B3)
                 => has_field_derivative(real,G,aa(real,real,G4,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z2)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B3)
                   => has_field_derivative(real,F2,aa(real,real,F9,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),C5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C5),B3)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3))),aa(real,real,G4,C5)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B3)),aa(real,real,G,A3))),aa(real,real,F9,C5)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_4976_summable__Leibniz_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(nat,real,A3,zero_zero(nat))),zero_zero(real))
         => ! [N3: nat] : member(real,suminf(real,aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)),one_one(nat)))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_4977_summable__Leibniz_I2_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(nat,real,A3,zero_zero(nat)))
         => ! [N3: nat] : member(real,suminf(real,aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),set_or1337092689740270186AtMost(real,aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_4978_summable__Leibniz_H_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2))
       => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2))
         => filterlim(nat,real,aTP_Lamp_qp(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_4979_trivial__limit__sequentially,axiom,
    at_top(nat) != bot_bot(filter(nat)) ).

% trivial_limit_sequentially
tff(fact_4980_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qq(A,fun(fun(nat,A),fun(nat,A)),C3),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_4981_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qr(A,fun(fun(nat,A),fun(nat,A)),C3),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_4982_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,A3: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qs(A,fun(fun(nat,A),fun(nat,A)),C3),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_4983_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),F3: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_qt(fun(nat,A),fun(nat,A),F2),F3,at_top(nat))
    <=> filterlim(nat,A,F2,F3,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_4984_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_4985_trivial__limit__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_top(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_top_linorder
tff(fact_4986_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ? [U3: fun(nat,A)] :
              ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(nat,A,U3,N3))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_4987_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ? [U3: fun(nat,A)] :
              ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,U3,N3)),Xa)
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_4988_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X4: fun(nat,A),Xa: A,A3: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( ? [N7: nat] :
              ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N2)),A3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A3) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_4989_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X4: fun(nat,A),Xa: A,A3: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( ? [N7: nat] :
              ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(nat,A,X4,N2)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),Xa) ) ) ) ).

% LIMSEQ_le_const
tff(fact_4990_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,N4: nat,C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(nat,A,F2,N2)) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),L) ) ) ) ).

% Lim_bounded2
tff(fact_4991_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),L: A,M5: nat,C2: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),N2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N2)),C2) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),C2) ) ) ) ).

% Lim_bounded
tff(fact_4992_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X4: fun(nat,A),Xa: A,Y3: fun(nat,A),Ya: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( filterlim(nat,A,Y3,topolo7230453075368039082e_nhds(A,Ya),at_top(nat))
           => ( ? [N7: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N2)),aa(nat,A,Y3,N2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ) ) ).

% LIMSEQ_le
tff(fact_4993_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N4: nat,X4: fun(nat,A),Y3: fun(nat,A),Xa: A,Ya: A] :
          ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N2)),aa(nat,A,Y3,N2)) )
         => ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
           => ( filterlim(nat,A,Y3,topolo7230453075368039082e_nhds(A,Ya),at_top(nat))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ) ) ).

% lim_mono
tff(fact_4994_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B3: fun(nat,A),S3: set(A),A3: A] :
          ( ! [N2: nat] : member(A,aa(nat,A,B3,N2),S3)
         => ( filterlim(nat,A,B3,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,complete_Sup_Sup(A),S3)) ) ) ) ).

% Sup_lim
tff(fact_4995_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B3: fun(nat,A),S3: set(A),A3: A] :
          ( ! [N2: nat] : member(A,aa(nat,A,B3,N2),S3)
         => ( filterlim(nat,A,B3,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S3)),A3) ) ) ) ).

% Inf_lim
tff(fact_4996_Inf__as__limit,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ? [U3: fun(nat,A)] :
              ( ! [N3: nat] : member(A,aa(nat,A,U3,N3),A4)
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,aa(set(A),A,complete_Inf_Inf(A),A4)),at_top(nat)) ) ) ) ).

% Inf_as_limit
tff(fact_4997_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_4998_mult__nat__right__at__top,axiom,
    ! [C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => filterlim(nat,nat,aTP_Lamp_qu(nat,fun(nat,nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_4999_mult__nat__left__at__top,axiom,
    ! [C3: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C3),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_5000_monoseq__convergent,axiom,
    ! [X4: fun(nat,real),B2: real] :
      ( topological_monoseq(real,X4)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,X4,I2))),B2)
       => ~ ! [L5: real] : ~ filterlim(nat,real,X4,topolo7230453075368039082e_nhds(real,L5),at_top(nat)) ) ) ).

% monoseq_convergent
tff(fact_5001_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: fun(nat,A),Xa: A] :
          ( topological_monoseq(A,A3)
         => ( filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
           => ( ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,N3)),Xa)
                & ! [M3: nat,N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,M3)),aa(nat,A,A3,N3)) ) )
              | ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(nat,A,A3,N3))
                & ! [M3: nat,N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A3,N3)),aa(nat,A,A3,M3)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_5002_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A] : filterlim(nat,A,aTP_Lamp_qv(A,fun(nat,A),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_5003_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_qw(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_5004_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X4: fun(nat,A),Xa: A,L: nat] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qx(fun(nat,A),fun(nat,fun(nat,A)),X4),L),topolo7230453075368039082e_nhds(A,Xa),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_5005_nested__sequence__unique,axiom,
    ! [F2: fun(nat,real),G: fun(nat,real)] :
      ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N2)),aa(nat,real,F2,aa(nat,nat,suc,N2)))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,G,aa(nat,nat,suc,N2))),aa(nat,real,G,N2))
       => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),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_qy(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L6: real] :
                ( ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N3)),L6)
                & filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L6),at_top(nat))
                & ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),aa(nat,real,G,N3))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L6),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_5006_LIMSEQ__inverse__zero,axiom,
    ! [X4: fun(nat,real)] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),N2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),R3),aa(nat,real,X4,N2)) )
     => filterlim(nat,real,aTP_Lamp_qz(fun(nat,real),fun(nat,real),X4),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_5007_LIMSEQ__root__const,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
     => filterlim(nat,real,aTP_Lamp_ra(real,fun(nat,real),C3),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_5008_increasing__LIMSEQ,axiom,
    ! [F2: fun(nat,real),L: real] :
      ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N2)),aa(nat,real,F2,aa(nat,nat,suc,N2)))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N2)),L)
       => ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F2,N3)),E)) )
         => filterlim(nat,real,F2,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_5009_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_rb(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_5010_LIMSEQ__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
       => filterlim(nat,real,power_power(real,Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_5011_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_rc(fun(nat,A),fun(nat,A),F2),aa(A,A,minus_minus(A,aa(nat,A,F2,zero_zero(nat))),C3)) ) ) ).

% telescope_sums'
tff(fact_5012_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),C3: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,C3),at_top(nat))
         => sums(A,aTP_Lamp_rd(fun(nat,A),fun(nat,A),F2),aa(A,A,minus_minus(A,C3),aa(nat,A,F2,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_5013_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xa: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_re(real,fun(real,fun(nat,real)),Xa),A3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_5014_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C3)),one_one(real))
     => filterlim(nat,real,power_power(real,C3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_5015_LIMSEQ__abs__realpow__zero,axiom,
    ! [C3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,abs_abs(real),C3)),one_one(real))
     => filterlim(nat,real,power_power(real,aa(real,real,abs_abs(real),C3)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_5016_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),Xa)
     => filterlim(nat,real,aTP_Lamp_rf(real,fun(nat,real),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_5017_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F2: fun(nat,A),S3: A] :
          ( sums(A,F2,S3)
        <=> filterlim(nat,A,aTP_Lamp_rg(fun(nat,A),fun(nat,A),F2),topolo7230453075368039082e_nhds(A,S3),at_top(nat)) ) ) ).

% sums_def'
tff(fact_5018_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),Xa: real] :
          ( filterlim(nat,real,aTP_Lamp_rh(fun(nat,A),fun(nat,real),F2),topolo7230453075368039082e_nhds(real,Xa),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),one_one(real))
           => summable(A,F2) ) ) ) ).

% root_test_convergence
tff(fact_5019_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),L4: A,R2: real] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N3)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X4,N3)),L4))),R2) ) ) ) ) ).

% LIMSEQ_D
tff(fact_5020_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),L4: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X4,N2)),L4))),R3) ) )
         => filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_5021_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),L4: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X4,N)),L4))),R5) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_5022_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => filterlim(nat,A,power_power(A,Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_5023_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F2: fun(A,nat),F3: filter(A),Xa: B] :
          ( filterlim(A,nat,F2,at_top(nat),F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,Xa)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_ri(fun(A,nat),fun(B,fun(A,B)),F2),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_power_zero
tff(fact_5024_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A)] :
          ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F2,N2))),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_5025_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),Df: A,Z: A,S3: fun(nat,A),A3: A] :
          ( has_field_derivative(A,F2,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N2: nat] : aa(nat,A,S3,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_rj(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F2),Z),S3),topolo7230453075368039082e_nhds(A,A3),at_top(nat))
               => ( Df = A3 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_5026_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,Xa)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_rk(A,fun(nat,A),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_5027_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xa))
         => filterlim(nat,A,aTP_Lamp_rl(A,fun(nat,A),Xa),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_5028_summable,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2))
       => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2))
         => summable(real,aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)) ) ) ) ).

% summable
tff(fact_5029_zeroseq__arctan__series,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xa)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_bp(real,fun(nat,real),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_5030_summable__Leibniz_H_I2_J,axiom,
    ! [A3: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2))
       => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)))),suminf(real,aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_5031_summable__Leibniz_H_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2))
       => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2))
         => filterlim(nat,real,aTP_Lamp_rm(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_5032_sums__alternating__upper__lower,axiom,
    ! [A3: fun(nat,real)] :
      ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2))
       => ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L6: real] :
              ( ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)))),L6)
              & filterlim(nat,real,aTP_Lamp_rm(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L6),at_top(nat))
              & ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L6),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_qp(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L6),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_5033_summable__Leibniz_H_I4_J,axiom,
    ! [A3: fun(nat,real),Na: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,A3,N2))
       => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,A3,aa(nat,nat,suc,N2))),aa(nat,real,A3,N2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3))),aa(set(nat),real,aa(fun(nat,real),fun(set(nat),real),groups7311177749621191930dd_sum(nat,real),aTP_Lamp_qo(fun(nat,real),fun(nat,real),A3)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_5034_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_rn(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_5035_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),D3: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F2,D3,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D3)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_ro(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F2),D3),Xa),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_5036_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xa: A,S3: set(A)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S3))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_rn(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% has_derivative_within
tff(fact_5037_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_ny(A,B)) ) ).

% bounded_linear_zero
tff(fact_5038_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
            ! [X2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K8)) ) ) ).

% bounded_linear.bounded
tff(fact_5039_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_rp(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_5040_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),K8)
              & ! [X2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K8)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_5041_has__derivative__within__singleton__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),G: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F2,G,topolo174197925503356063within(A,Xa,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))
        <=> real_V3181309239436604168linear(A,B,G) ) ) ).

% has_derivative_within_singleton_iff
tff(fact_5042_filterlim__pow__at__top,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,real),F3: filter(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pb(nat,fun(fun(A,real),fun(A,real)),Na),F2),at_top(real),F3) ) ) ).

% filterlim_pow_at_top
tff(fact_5043_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F2)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [X2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F2,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K8)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_5044_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),K4: real] :
          ( ! [X3: A,Y: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,X3)),aa(A,B,F2,Y))
         => ( ! [R3: real,X3: A] : aa(A,B,F2,aa(A,A,real_V8093663219630862766scaleR(A,R3),X3)) = aa(B,B,real_V8093663219630862766scaleR(B,R3),aa(A,B,F2,X3))
           => ( ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),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)),K4))
             => real_V3181309239436604168linear(A,B,F2) ) ) ) ) ).

% bounded_linear_intro
tff(fact_5045_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),C3: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C3),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_5046_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F2: fun(A,real),C3: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C3),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rr(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_5047_tendsto__neg__powr,axiom,
    ! [A: $tType,S3: real,F2: fun(A,real),F3: filter(A)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),S3),zero_zero(real))
     => ( filterlim(A,real,F2,at_top(real),F3)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rs(real,fun(fun(A,real),fun(A,real)),S3),F2),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ).

% tendsto_neg_powr
tff(fact_5048_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B3: real,F2: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),X3)
         => ? [Y2: real] :
              ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),zero_zero(real)) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,B3)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_5049_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F9: fun(A,B),Xa: A,F2: fun(A,B),S3: set(A)] :
          ( real_V3181309239436604168linear(A,B,F9)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_rt(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F9),Xa),F2),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,S3))
           => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% has_derivativeI
tff(fact_5050_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xa: A,S3: set(A)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S3))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_ru(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F2),F9),Xa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% has_derivative_at_within
tff(fact_5051_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xa: A] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & ? [E3: fun(A,B)] :
                ( ! [H5: A] : aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xa)),aa(A,B,F9,H5))),aa(A,B,E3,H5))
                & filterlim(A,real,aTP_Lamp_rv(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_5052_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),F3: filter(A)] :
          ( has_derivative(A,B,F2,F9,F3)
        <=> ( real_V3181309239436604168linear(A,B,F9)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_rx(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),F2),F9),F3),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% has_derivative_def
tff(fact_5053_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xa: A,S: set(A),F2: fun(A,B),F9: fun(A,B)] :
          ( member(A,Xa,S)
         => ( topolo1002775350975398744n_open(A,S)
           => ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S))
            <=> ( real_V3181309239436604168linear(A,B,F9)
                & ? [E3: fun(A,B)] :
                    ( ! [H5: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H5),S)
                       => ( aa(A,B,F2,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F2,Xa)),aa(A,B,F9,H5))),aa(A,B,E3,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_rv(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_5054_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F9: fun(A,B),S3: set(A),Xa: A,F2: fun(A,B),H6: fun(A,real)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
         => ( real_V3181309239436604168linear(A,B,F9)
           => ( ! [Y: A] :
                  ( member(A,Y,S3)
                 => ( ( Y != Xa )
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y,Xa)),E2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),divide_divide(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,F2,Y)),aa(A,B,F2,Xa))),aa(A,B,F9,aa(A,A,minus_minus(A,Y),Xa)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Xa)))),aa(A,real,H6,Y)) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xa,S3))
               => has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S3)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_5055_open__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1002775350975398744n_open(A,bot_bot(set(A))) ) ).

% open_empty
tff(fact_5056_open__Un,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T3: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
         => ( topolo1002775350975398744n_open(A,T3)
           => topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) ) ) ) ).

% open_Un
tff(fact_5057_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : real_V557655796197034286t_dist(A,zero_zero(A),Xa) = real_V7770717601297561774m_norm(A,Xa) ) ).

% dist_0_norm
tff(fact_5058_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Ya: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xa,Ya))
        <=> ( Xa != Ya ) ) ) ).

% zero_less_dist_iff
tff(fact_5059_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Ya: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Ya)),zero_zero(real))
        <=> ( Xa = Ya ) ) ) ).

% dist_le_zero_iff
tff(fact_5060_open__INT,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A4: set(A),B2: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => topolo1002775350975398744n_open(B,aa(A,set(B),B2,X3)) )
           => topolo1002775350975398744n_open(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))) ) ) ) ).

% open_INT
tff(fact_5061_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
        <=> ! [X: A] :
              ( member(A,X,S)
             => ? [E3: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
                  & ! [Y4: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Y4,X)),E3)
                     => member(A,Y4,S) ) ) ) ) ) ).

% open_dist
tff(fact_5062_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Ya: A,Xa: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Ya,Xa)),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Ya)),E2) ) ) ).

% dist_commute_lessI
tff(fact_5063_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,D2: real] : topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_ry(A,fun(real,fun(A,$o)),Xa),D2))) ) ).

% open_ball
tff(fact_5064_openI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( ! [X3: A] :
              ( member(A,X3,S)
             => ? [T8: set(A)] :
                  ( topolo1002775350975398744n_open(A,T8)
                  & member(A,X3,T8)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T8),S) ) )
         => topolo1002775350975398744n_open(A,S) ) ) ).

% openI
tff(fact_5065_open__subopen,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo1002775350975398744n_open(A,S)
        <=> ! [X: A] :
              ( member(A,X,S)
             => ? [T9: set(A)] :
                  ( topolo1002775350975398744n_open(A,T9)
                  & member(A,X,T9)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),S) ) ) ) ) ).

% open_subopen
tff(fact_5066_first__countable__basis,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Xa: A] :
        ? [A7: fun(nat,set(A))] :
          ( ! [I3: nat] :
              ( member(A,Xa,aa(nat,set(A),A7,I3))
              & topolo1002775350975398744n_open(A,aa(nat,set(A),A7,I3)) )
          & ! [S9: set(A)] :
              ( ( topolo1002775350975398744n_open(A,S9)
                & member(A,Xa,S9) )
             => ? [I2: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),A7,I2)),S9) ) ) ) ).

% first_countable_basis
tff(fact_5067_norm__conv__dist,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xa: A] : real_V7770717601297561774m_norm(A,Xa) = real_V557655796197034286t_dist(A,Xa,zero_zero(A)) ) ).

% norm_conv_dist
tff(fact_5068_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa != Ya )
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xa,Ya)) ) ) ).

% dist_pos_lt
tff(fact_5069_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Ya: A] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Ya)),zero_zero(real)) ) ).

% dist_not_less_zero
tff(fact_5070_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Ya: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V557655796197034286t_dist(A,Xa,Ya)) ) ).

% zero_le_dist
tff(fact_5071_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A4: set(A),Xa: A] :
          ( topolo1002775350975398744n_open(A,A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Xa) )
           => ~ member(A,aa(set(A),A,complete_Sup_Sup(A),A4),A4) ) ) ) ).

% Sup_notin_open
tff(fact_5072_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A4: set(A),Xa: A] :
          ( topolo1002775350975398744n_open(A,A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X3) )
           => ~ member(A,aa(set(A),A,complete_Inf_Inf(A),A4),A4) ) ) ) ).

% Inf_notin_open
tff(fact_5073_not__open__singleton,axiom,
    ! [A: $tType] :
      ( topolo8386298272705272623_space(A)
     => ! [Xa: A] : ~ topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ) ).

% not_open_singleton
tff(fact_5074_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Ya: A,E1: real,X23: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,Ya)),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,Ya)),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X23)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% dist_triangle_less_add
tff(fact_5075_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Z: A,Ya: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Z)),real_V557655796197034286t_dist(A,Ya,Z))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa,Ya)),E2) ) ) ).

% dist_triangle_lt
tff(fact_5076_separation__t2,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa != Ya )
        <=> ? [U4: set(A),V4: set(A)] :
              ( topolo1002775350975398744n_open(A,U4)
              & topolo1002775350975398744n_open(A,V4)
              & member(A,Xa,U4)
              & member(A,Ya,V4)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U4),V4) = bot_bot(set(A)) ) ) ) ) ).

% separation_t2
tff(fact_5077_hausdorff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xa: A,Ya: A] :
          ( ( Xa != Ya )
         => ? [U5: set(A),V5: set(A)] :
              ( topolo1002775350975398744n_open(A,U5)
              & topolo1002775350975398744n_open(A,V5)
              & member(A,Xa,U5)
              & member(A,Ya,V5)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V5) = bot_bot(set(A)) ) ) ) ) ).

% hausdorff
tff(fact_5078_dist__triangle,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Z: A,Ya: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Ya)),real_V557655796197034286t_dist(A,Ya,Z))) ) ).

% dist_triangle
tff(fact_5079_dist__triangle2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Ya)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Z)),real_V557655796197034286t_dist(A,Ya,Z))) ) ).

% dist_triangle2
tff(fact_5080_dist__triangle3,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Ya: A,A3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Ya)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,A3,Xa)),real_V557655796197034286t_dist(A,A3,Ya))) ) ).

% dist_triangle3
tff(fact_5081_dist__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A,Z: A,Ya: A,E2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,Xa,Z)),real_V557655796197034286t_dist(A,Ya,Z))),E2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,Xa,Ya)),E2) ) ) ).

% dist_triangle_le
tff(fact_5082_at__within__open__subset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A,S: set(A),T3: set(A)] :
          ( member(A,A3,S)
         => ( topolo1002775350975398744n_open(A,S)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
             => ( topolo174197925503356063within(A,A3,T3) = topolo174197925503356063within(A,A3,top_top(set(A))) ) ) ) ) ) ).

% at_within_open_subset
tff(fact_5083_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),Xa: A,Ya: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( member(A,Xa,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
             => ? [B5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B5)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,Xa,B5)),S) ) ) ) ) ) ).

% open_right
tff(fact_5084_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A3: A,B3: A,C3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V557655796197034286t_dist(A,A3,B3)),real_V557655796197034286t_dist(A,B3,C3)))),real_V557655796197034286t_dist(A,A3,C3)) ) ).

% abs_dist_diff_le
tff(fact_5085_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),F9: A,A3: A,S: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F2,F9,topolo174197925503356063within(A,A3,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,A3,S)
             => ( ! [X3: A] :
                    ( member(A,X3,S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A3)),D2)
                     => ( aa(A,A,F2,X3) = aa(A,A,G,X3) ) ) )
               => has_field_derivative(A,G,F9,topolo174197925503356063within(A,A3,S)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_5086_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F9: fun(A,B),Xa: A,S3: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F2,F9,topolo174197925503356063within(A,Xa,S3))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( member(A,Xa,S3)
             => ( ! [X7: A] :
                    ( member(A,X7,S3)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X7,Xa)),D2)
                     => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) )
               => has_derivative(A,B,G,F9,topolo174197925503356063within(A,Xa,S3)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_5087_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X4)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M8: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M2),aa(nat,A,X4,N))),E3) ) ) ) ) ) ).

% Cauchy_def
tff(fact_5088_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S3: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S3)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [N6: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S3,N),aa(nat,A,S3,N6))),E3) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_5089_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X4)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => ? [M9: nat] :
              ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),M3)
               => ! [N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M9),N3)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M3),aa(nat,A,X4,N3))),E2) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_5090_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M4),aa(nat,A,X4,N2))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X4) ) ) ).

% metric_CauchyI
tff(fact_5091_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F2: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F2,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S10: set(A)] :
              ( topolo1002775350975398744n_open(A,S10)
             => ( member(A,F0,S10)
               => ? [N6: nat] :
                  ! [N: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                   => member(A,aa(nat,A,F2,N),S10) ) ) ) ) ) ).

% lim_explicit
tff(fact_5092_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_rz(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F3,aa(fun(A,B),fun(A,B),aTP_Lamp_qf(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_divide
tff(fact_5093_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_rz(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_qg(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_inverse
tff(fact_5094_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_rz(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_qh(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_sgn
tff(fact_5095_metric__LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L: B,A3: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A3 )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,G,X3),M)),real_V557655796197034286t_dist(B,aa(A,B,F2,X3),L)) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ).

% metric_LIM_imp_LIM
tff(fact_5096_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),L: B,Xa: A,S: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xa,S))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
           => ( ! [X7: A] :
                  ( member(A,X7,S)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X7,Xa))
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X7,Xa)),D2)
                     => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,Xa,S)) ) ) ) ) ).

% Lim_transform_within
tff(fact_5097_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Ya: A,E2: real,X23: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,Ya)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,Ya)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X23)),E2) ) ) ) ).

% dist_triangle_half_l
tff(fact_5098_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Ya: A,X15: A,E2: real,X23: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Ya,X15)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit0(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Ya,X23)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit0(one2))))
           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X23)),E2) ) ) ) ).

% dist_triangle_half_r
tff(fact_5099_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,X23: A,E2: real,X33: A,X42: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X23)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit1(one2))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X23,X33)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit1(one2))))
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X33,X42)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),bit1(one2))))
             => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X15,X42)),E2) ) ) ) ) ).

% dist_triangle_third
tff(fact_5100_at__within__nhd,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A,S: set(A),T3: set(A),U2: set(A)] :
          ( member(A,Xa,S)
         => ( topolo1002775350975398744n_open(A,S)
           => ( ( aa(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)),T3),S)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = aa(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)),U2),S)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) )
             => ( topolo174197925503356063within(A,Xa,T3) = topolo174197925503356063within(A,Xa,U2) ) ) ) ) ) ).

% at_within_nhd
tff(fact_5101_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G2: filter(B),Xa: A,S: set(A),F3: filter(B),D2: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,G2,topolo174197925503356063within(A,Xa,S))
         => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G2),F3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D2)
             => ( ! [X7: A] :
                    ( member(A,X7,S)
                   => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X7,Xa))
                     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X7,Xa)),D2)
                       => ( aa(A,B,F2,X7) = aa(A,B,G,X7) ) ) ) )
               => filterlim(A,B,F2,F3,topolo174197925503356063within(A,Xa,S)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_5102_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => ? [M10: nat] :
                ! [M4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M4)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),N2)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M4),aa(nat,A,X4,N2))),E) ) ) )
         => topolo3814608138187158403Cauchy(A,X4) ) ) ).

% CauchyI'
tff(fact_5103_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F2: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [M8: nat] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
                 => ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F2,M2),aa(nat,A,F2,N))),E3) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_5104_at__eq__bot__iff,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A] :
          ( ( topolo174197925503356063within(A,A3,top_top(set(A))) = bot_bot(filter(A)) )
        <=> topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ).

% at_eq_bot_iff
tff(fact_5105_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A3: A,R: real,F2: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R)
           => ( ! [X3: A] :
                  ( ( X3 != A3 )
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A3)),R)
                   => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
             => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_5106_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A3: A,F2: fun(A,B),L4: B] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [S8: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S8)
                  & ! [X3: A] :
                      ( ( ( X3 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A3)),S8) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X3),L4)),R3) ) ) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_5107_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L4: B,A3: A,R2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [S4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S4)
                & ! [X2: A] :
                    ( ( ( X2 != A3 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X2,A3)),S4) )
                   => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X2),L4)),R2) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_5108_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),L4: B,A3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A] :
                      ( ( ( X != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A3)),S7) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X),L4)),R5) ) ) ) ) ) ).

% LIM_def
tff(fact_5109_lim__sequentially,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),L4: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,N),L4)),R5) ) ) ) ) ).

% lim_sequentially
tff(fact_5110_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),L4: A] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No2),N2)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,N2),L4)),R3) ) )
         => filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_5111_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),L4: A,R2: real] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => ? [No: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No),N3)
               => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,N3),L4)),R2) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_5112_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X4)
        <=> ! [J3: nat] :
            ? [M8: nat] :
            ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),M2)
             => ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M8),N)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,M2),aa(nat,A,X4,N))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_5113_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F2: fun(A,B),B3: B,A3: A,G: fun(B,C),C3: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,B3),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(B,B3,top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X3: A] :
                      ( ( ( X3 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A3)),D4) )
                     => ( aa(A,B,F2,X3) != B3 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_sa(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,C3),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_5114_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_sb(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_pf(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_tan
tff(fact_5115_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_sb(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F3,aTP_Lamp_pw(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_cot
tff(fact_5116_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F3: filter(A),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( cosh(B,aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_rz(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aTP_Lamp_ql(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_tanh
tff(fact_5117_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_rz(A,A))))
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_sc(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh
tff(fact_5118_metric__isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,F2: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),F2)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F2,A3),top_top(set(B))))
           => ( ? [D4: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D4)
                  & ! [X3: A] :
                      ( ( ( X3 != A3 )
                        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X3,A3)),D4) )
                     => ( aa(A,B,F2,X3) != aa(A,B,F2,A3) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_sa(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_5119_tendsto__offset__zero__iff,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C)
        & zero(A) )
     => ! [A3: B,S: set(B),F2: fun(B,C),L4: C] :
          ( nO_MATCH(A,B,zero_zero(A),A3)
         => ( member(B,A3,S)
           => ( topolo1002775350975398744n_open(B,S)
             => ( filterlim(B,C,F2,topolo7230453075368039082e_nhds(C,L4),topolo174197925503356063within(B,A3,S))
              <=> filterlim(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_qc(B,fun(fun(B,C),fun(B,C)),A3),F2),topolo7230453075368039082e_nhds(C,L4),topolo174197925503356063within(B,zero_zero(B),top_top(set(B)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_5120_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( topolo3448309680560233919inuous(A,real,F3,G)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_rz(A,A))))
             => ( ( aa(A,real,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_rz(A,A))) != one_one(real) )
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_rz(A,A))))
                 => topolo3448309680560233919inuous(A,real,F3,aa(fun(A,real),fun(A,real),aTP_Lamp_qn(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_5121_LIMSEQ__iff__nz,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X4: fun(nat,A),L4: A] :
          ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [No3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),No3)
                  & ! [N: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),No3),N)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X4,N),L4)),R5) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_5122_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: set(A)] :
          ( topolo6688025880775521714ounded(A,S)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [K3: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),K3)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image2(A,set(A),aTP_Lamp_se(real,fun(A,set(A)),E3)),K3))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_5123_filterlim__pow__at__bot__even,axiom,
    ! [Na: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_sf(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_top(real),F3) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_5124_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_sg(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_5125_totally__bounded__empty,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => topolo6688025880775521714ounded(A,bot_bot(set(A))) ) ).

% totally_bounded_empty
tff(fact_5126_at__top__le__at__infinity,axiom,
    aa(filter(real),$o,aa(filter(real),fun(filter(real),$o),ord_less_eq(filter(real)),at_top(real)),at_infinity(real)) ).

% at_top_le_at_infinity
tff(fact_5127_at__bot__le__at__infinity,axiom,
    aa(filter(real),$o,aa(filter(real),fun(filter(real),$o),ord_less_eq(filter(real)),at_bot(real)),at_infinity(real)) ).

% at_bot_le_at_infinity
tff(fact_5128_trivial__limit__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( at_bot(A) != bot_bot(filter(A)) ) ) ).

% trivial_limit_at_bot_linorder
tff(fact_5129_totally__bounded__subset,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S: set(A),T3: set(A)] :
          ( topolo6688025880775521714ounded(A,S)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S)
           => topolo6688025880775521714ounded(A,T3) ) ) ) ).

% totally_bounded_subset
tff(fact_5130_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_5131_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [F2: fun(A,B),C3: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C3),F3)
         => ( ( C3 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F3)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sh(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),at_infinity(B),F3) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_5132_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),C3: B,F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C3),F3)
         => ( filterlim(A,B,G,at_infinity(B),F3)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_si(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F3) ) ) ) ).

% tendsto_divide_0
tff(fact_5133_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),F3: filter(A),Na: nat] :
          ( filterlim(A,B,F2,at_infinity(B),F3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_sj(fun(A,B),fun(nat,fun(A,B)),F2),Na),at_infinity(B),F3) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_5134_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C3: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C3),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),C3)
       => ( filterlim(A,real,G,at_bot(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_5135_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_5136_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_pl(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_5137_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),C3: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C3),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),zero_zero(real))
       => ( filterlim(A,real,G,at_top(real),F3)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rq(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_bot(real),F3) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_5138_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),C3: A,F3: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F2,topolo7230453075368039082e_nhds(A,C3),F3)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F3)
           => ( ( C3 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_my(fun(A,A),fun(fun(A,A),fun(A,A)),F2),G),at_infinity(A),F3) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_5139_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xa: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,Xa))
         => filterlim(nat,A,power_power(A,Xa),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_5140_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B3: real,F2: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3)
         => ? [Y2: real] :
              ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2) ) )
     => ( filterlim(real,real,F2,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less(real),Flim),aa(real,real,F2,B3)) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_5141_filterlim__pow__at__bot__odd,axiom,
    ! [Na: nat,F2: fun(real,real),F3: filter(real)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( filterlim(real,real,F2,at_bot(real),F3)
       => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_sf(nat,fun(fun(real,real),fun(real,real)),Na),F2),at_bot(real),F3) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_5142_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C3: fun(nat,A),K: nat,Na: nat,B2: real] :
          ( ( aa(nat,A,C3,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Na)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_sk(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C3),Na),B2)),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_5143_GMVT,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),G: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ! [X3: real] :
            ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F2) )
       => ( ! [X3: real] :
              ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3) )
             => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) )
         => ( ! [X3: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),X3)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),B3) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),G) )
           => ( ! [X3: real] :
                  ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
                    & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X3,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C5: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C5,top_top(set(real))))
                  & has_field_derivative(real,F2,F_c,topolo174197925503356063within(real,C5,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),C5)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),C5),B3)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B3)),aa(real,real,G,A3))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_5144_Bfun__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( ( A3 != zero_zero(B) )
           => bfun(A,B,aTP_Lamp_pl(fun(A,B),fun(A,B),F2),F3) ) ) ) ).

% Bfun_inverse
tff(fact_5145_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_ic(fun(nat,$o),fun(nat,$o),P)),at_top(nat))
    <=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_5146_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(nat,fun(nat,$o),aTP_Lamp_sl(fun(nat,$o),fun(nat,fun(nat,$o)),P),K)),at_top(nat))
    <=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_5147_eventually__top,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),top_top(filter(A)))
    <=> ! [X_13: A] : aa(A,$o,P,X_13) ) ).

% eventually_top
tff(fact_5148_eventually__const,axiom,
    ! [A: $tType,F3: filter(A),P: $o] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_mb($o,fun(A,$o),(P))),F3)
      <=> (P) ) ) ).

% eventually_const
tff(fact_5149_differentiable__cmult__right__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Q5: fun(A,B),C3: B,Ta: A] :
          ( differentiable(A,B,aa(B,fun(A,B),aTP_Lamp_sm(fun(A,B),fun(B,fun(A,B)),Q5),C3),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C3 = zero_zero(B) )
            | differentiable(A,B,Q5,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_right_iff
tff(fact_5150_differentiable__cmult__left__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [C3: B,Q5: fun(A,B),Ta: A] :
          ( differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sn(B,fun(fun(A,B),fun(A,B)),C3),Q5),topolo174197925503356063within(A,Ta,top_top(set(A))))
        <=> ( ( C3 = zero_zero(B) )
            | differentiable(A,B,Q5,topolo174197925503356063within(A,Ta,top_top(set(A)))) ) ) ) ).

% differentiable_cmult_left_iff
tff(fact_5151_eventually__at__bot__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [C3: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_so(A,fun(A,$o),C3)),at_bot(A)) ) ).

% eventually_at_bot_not_equal
tff(fact_5152_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F22: filter(B),F1: filter(A)] :
      ( filterlim(A,B,F2,F22,F1)
    <=> ! [P5: fun(B,$o)] :
          ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P5),F22)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_sp(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P5)),F1) ) ) ).

% filterlim_iff
tff(fact_5153_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F1: filter(A),F12: filter(A),F22: filter(B),F23: filter(B),F2: fun(B,A),G: fun(B,A)] :
      ( ( F1 = F12 )
     => ( ( F22 = F23 )
       => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_sq(fun(B,A),fun(fun(B,A),fun(B,$o)),F2),G)),F22)
         => ( filterlim(B,A,F2,F1,F22)
          <=> filterlim(B,A,G,F12,F23) ) ) ) ) ).

% filterlim_cong
tff(fact_5154_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: filter(A),F2: fun(B,A),G2: filter(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ( filterlim(B,A,F2,F3,G2)
       => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_sr(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2)),G2) ) ) ).

% eventually_compose_filterlim
tff(fact_5155_trivial__limit__def,axiom,
    ! [A: $tType,F3: filter(A)] :
      ( ( F3 = bot_bot(filter(A)) )
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ao(A,$o)),F3) ) ).

% trivial_limit_def
tff(fact_5156_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_mb($o,fun(A,$o),(P))),F3)
    <=> ( (P)
        | ( F3 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_5157_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X3: A] : ~ aa(A,$o,P,X3)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),Net) ) ) ).

% False_imp_not_eventually
tff(fact_5158_eventually__happens_H,axiom,
    ! [A: $tType,F3: filter(A),P: fun(A,$o)] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
       => ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).

% eventually_happens'
tff(fact_5159_eventually__happens,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),Net)
     => ( ( Net = bot_bot(filter(A)) )
        | ? [X_1: A] : aa(A,$o,P,X_1) ) ) ).

% eventually_happens
tff(fact_5160_eventually__bot,axiom,
    ! [A: $tType,P: fun(A,$o)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),bot_bot(filter(A))) ).

% eventually_bot
tff(fact_5161_eventuallyI,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( ! [X3: A] : aa(A,$o,P,X3)
     => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3) ) ).

% eventuallyI
tff(fact_5162_filter__eq__iff,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A)] :
      ( ( F3 = F8 )
    <=> ! [P5: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P5),F3)
        <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P5),F8) ) ) ).

% filter_eq_iff
tff(fact_5163_eventually__mono,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ( ! [X3: A] :
            ( aa(A,$o,P,X3)
           => aa(A,$o,Q,X3) )
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F3) ) ) ).

% eventually_mono
tff(fact_5164_not__eventuallyD,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ? [X3: A] : ~ aa(A,$o,P,X3) ) ).

% not_eventuallyD
tff(fact_5165_always__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( ! [X_1: A] : aa(A,$o,P,X_1)
     => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3) ) ).

% always_eventually
tff(fact_5166_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F3)
       => ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3) ) ) ).

% not_eventually_impI
tff(fact_5167_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F3) ) ) ).

% eventually_conj_iff
tff(fact_5168_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F3) ) ) ).

% eventually_rev_mp
tff(fact_5169_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ss(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
      <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F3) ) ) ).

% eventually_subst
tff(fact_5170_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o),R: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F3)
       => ( ! [I2: A] :
              ( aa(A,$o,P,I2)
             => ( aa(A,$o,Q,I2)
               => aa(A,$o,R,I2) ) )
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R),F3) ) ) ) ).

% eventually_elim2
tff(fact_5171_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3) ) ) ).

% eventually_conj
tff(fact_5172_eventually__True,axiom,
    ! [A: $tType,F3: filter(A)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_mf(A,$o)),F3) ).

% eventually_True
tff(fact_5173_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),F3) ) ) ).

% eventually_mp
tff(fact_5174_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,$o),C2: $o,F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa($o,fun(A,$o),aTP_Lamp_st(fun(A,$o),fun($o,fun(A,$o)),P),(C2))),F3)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
        | (C2) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_5175_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C2: $o,P: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_su($o,fun(fun(A,$o),fun(A,$o)),(C2)),P)),F3)
    <=> ( (C2)
        | aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_5176_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C2: $o,P: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_sv($o,fun(fun(A,$o),fun(A,$o)),(C2)),P)),F3)
    <=> ( (C2)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_5177_eventually__inf,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),F8: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),F8))
    <=> ? [Q7: fun(A,$o),R6: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q7),F3)
          & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R6),F8)
          & ! [X: A] :
              ( ( aa(A,$o,Q7,X)
                & aa(A,$o,R6,X) )
             => aa(A,$o,P,X) ) ) ) ).

% eventually_inf
tff(fact_5178_filter__leD,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A),P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F8)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F8)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3) ) ) ).

% filter_leD
tff(fact_5179_filter__leI,axiom,
    ! [A: $tType,F8: filter(A),F3: filter(A)] :
      ( ! [P6: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P6),F8)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P6),F3) )
     => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F8) ) ).

% filter_leI
tff(fact_5180_le__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F8)
    <=> ! [P5: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P5),F8)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P5),F3) ) ) ).

% le_filter_def
tff(fact_5181_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),K4: real,F3: filter(A)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aTP_Lamp_sw(fun(A,B),fun(real,fun(A,$o)),F2),K4)),F3)
         => bfun(A,B,F2,F3) ) ) ).

% BfunI
tff(fact_5182_eventually__Sup,axiom,
    ! [A: $tType,P: fun(A,$o),S: set(filter(A))] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S))
    <=> ! [X: filter(A)] :
          ( member(filter(A),X,S)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),X) ) ) ).

% eventually_Sup
tff(fact_5183_eventually__sup,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),F8: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F3),F8))
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F8) ) ) ).

% eventually_sup
tff(fact_5184_Bseq__eventually__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(nat,A),G: fun(nat,B)] :
          ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(fun(nat,B),fun(nat,$o),aTP_Lamp_sx(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),F2),G)),at_top(nat))
         => ( bfun(nat,B,G,at_top(nat))
           => bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_eventually_mono
tff(fact_5185_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C3: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_sy(A,fun(A,$o),C3)),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_5186_eventually__False__sequentially,axiom,
    ~ aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_sz(nat,$o)),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_5187_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_top(A))
        <=> ? [N6: A] :
            ! [N: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N6),N)
             => aa(A,$o,P,N) ) ) ) ).

% eventually_at_top_linorder
tff(fact_5188_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A,P: fun(A,$o)] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),X3)
             => aa(A,$o,P,X3) )
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_5189_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_top(A))
        <=> ? [N6: A] :
            ! [N: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N6),N)
             => aa(A,$o,P,N) ) ) ) ).

% eventually_at_top_dense
tff(fact_5190_eventually__sequentially,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),at_top(nat))
    <=> ? [N6: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
         => aa(nat,$o,P,N) ) ) ).

% eventually_sequentially
tff(fact_5191_eventually__sequentiallyI,axiom,
    ! [C3: nat,P: fun(nat,$o)] :
      ( ! [X3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C3),X3)
         => aa(nat,$o,P,X3) )
     => aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_5192_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_bot(A))
        <=> ? [N6: A] :
            ! [N: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),N),N6)
             => aa(A,$o,P,N) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_5193_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_bot(A))
        <=> ? [N6: A] :
            ! [N: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),N),N6)
             => aa(A,$o,P,N) ) ) ) ).

% eventually_at_bot_dense
tff(fact_5194_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),ord_less_eq(A),C3)),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_5195_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C3: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),ord_less(A),C3)),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_5196_differentiable__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),Xa: A,S3: set(A),Ta: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xa,S3))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S3)
           => differentiable(A,B,F2,topolo174197925503356063within(A,Xa,Ta)) ) ) ) ).

% differentiable_within_subset
tff(fact_5197_le__sequentially,axiom,
    ! [F3: filter(nat)] :
      ( aa(filter(nat),$o,aa(filter(nat),fun(filter(nat),$o),ord_less_eq(filter(nat)),F3),at_top(nat))
    <=> ! [N6: nat] : aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),N6)),F3) ) ).

% le_sequentially
tff(fact_5198_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),aTP_Lamp_ta(A,fun(A,$o)),C3)),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_5199_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C3: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_tb(A,fun(A,$o),C3)),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_5200_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(B),G2: filter(A),F8: filter(B),G5: filter(A),F9: fun(A,B)] :
      ( filterlim(A,B,F2,F3,G2)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F3),F8)
       => ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G5),G2)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_tc(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),F9)),G5)
           => filterlim(A,B,F9,F8,G5) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_5201_eventually__INF1,axiom,
    ! [B: $tType,A: $tType,I: A,I5: set(A),P: fun(B,$o),F3: fun(A,filter(B))] :
      ( member(A,I,I5)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(A,filter(B),F3,I))
       => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F3),I5))) ) ) ).

% eventually_INF1
tff(fact_5202_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
         => ~ ! [B4: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B4)
               => ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aTP_Lamp_sw(fun(A,B),fun(real,fun(A,$o)),F2),B4)),F3) ) ) ) ).

% BfunE
tff(fact_5203_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aTP_Lamp_sw(fun(A,B),fun(real,fun(A,$o)),F2),K5)),F3) ) ) ) ).

% Bfun_def
tff(fact_5204_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( bfun(A,B,F2,F3)
        <=> ? [Y4: B,K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_td(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),Y4),K5)),F3) ) ) ) ).

% Bfun_metric_def
tff(fact_5205_differentiable__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [S3: set(A),F2: fun(A,fun(B,C)),Net: filter(B)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => ( ! [X3: A] :
                ( member(A,X3,S3)
               => differentiable(B,C,aa(A,fun(B,C),F2,X3),Net) )
           => differentiable(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tf(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S3),F2),Net) ) ) ) ).

% differentiable_sum
tff(fact_5206_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B3: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),top_top(A))
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),top_top(A))
                & ! [Z5: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Z5)
                   => aa(A,$o,P,Z5) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_5207_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P,X3)
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P,X3)
                 => aa(A,$o,Q,aa(B,A,G,X3)) )
             => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),at_top(A))
               => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),at_top(B))
                 => filterlim(A,B,F2,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_5208_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xa: A] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,Xa,aa(A,set(A),set_ord_lessThan(A),Xa)))
        <=> ? [B7: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Xa)
              & ! [Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Y4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xa)
                   => aa(A,$o,P,Y4) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_5209_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Ya: A,Xa: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,Xa,aa(A,set(A),set_ord_lessThan(A),Xa)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Xa)
                & ! [Y4: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B7),Y4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),Xa)
                     => aa(A,$o,P,Y4) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_5210_eventually__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_infinity(A))
        <=> ? [B7: real] :
            ! [X: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X))
             => aa(A,$o,P,X) ) ) ) ).

% eventually_at_infinity
tff(fact_5211_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),G: fun(A,B),Net: filter(A),H: fun(A,B),C3: B] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_tg(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G)),Net)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_tg(fun(A,B),fun(fun(A,B),fun(A,$o)),G),H)),Net)
           => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C3),Net)
             => ( filterlim(A,B,H,topolo7230453075368039082e_nhds(B,C3),Net)
               => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C3),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_5212_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Xa: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xa),F3)
        <=> ( ! [L3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L3),Xa)
               => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_th(fun(A,B),fun(B,fun(A,$o)),F2),L3)),F3) )
            & ! [U6: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Xa),U6)
               => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ti(fun(A,B),fun(B,fun(A,$o)),F2),U6)),F3) ) ) ) ) ).

% order_tendsto_iff
tff(fact_5213_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Ya: A,F2: fun(B,A),F3: filter(B)] :
          ( ! [A5: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),Ya)
             => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_tj(fun(B,A),fun(A,fun(B,$o)),F2),A5)),F3) )
         => ( ! [A5: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),A5)
               => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_tk(fun(B,A),fun(A,fun(B,$o)),F2),A5)),F3) )
           => filterlim(B,A,F2,topolo7230453075368039082e_nhds(A,Ya),F3) ) ) ) ).

% order_tendstoI
tff(fact_5214_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Ya: B,F3: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Ya),F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),Ya)
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_th(fun(A,B),fun(B,fun(A,$o)),F2),A3)),F3) ) ) ) ).

% order_tendstoD(1)
tff(fact_5215_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),Ya: B,F3: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Ya),F3)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Ya),A3)
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ti(fun(A,B),fun(B,fun(A,$o)),F2),A3)),F3) ) ) ) ).

% order_tendstoD(2)
tff(fact_5216_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),F3) ) ) ).

% filterlim_at_top
tff(fact_5217_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C3: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),C3),Z7)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),F3) ) ) ) ).

% filterlim_at_top_ge
tff(fact_5218_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F2,at_top(B),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G)),F3)
           => filterlim(A,B,G,at_top(B),F3) ) ) ) ).

% filterlim_at_top_mono
tff(fact_5219_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_tn(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),F3) ) ) ).

% filterlim_at_top_dense
tff(fact_5220_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_to(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),F3) ) ) ).

% filterlim_at_bot
tff(fact_5221_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),F3: filter(A),C3: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Z7),C3)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_to(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),F3) ) ) ) ).

% filterlim_at_bot_le
tff(fact_5222_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),F3) ) ) ).

% filterlim_at_bot_dense
tff(fact_5223_BseqI_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A),K4: real] :
          ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N2))),K4)
         => bfun(nat,A,X4,at_top(nat)) ) ) ).

% BseqI'
tff(fact_5224_real__tendsto__sandwich,axiom,
    ! [A: $tType,F2: fun(A,real),G: fun(A,real),Net: filter(A),H: fun(A,real),C3: real] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,real),fun(A,$o),aTP_Lamp_tq(fun(A,real),fun(fun(A,real),fun(A,$o)),F2),G)),Net)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,real),fun(A,$o),aTP_Lamp_tq(fun(A,real),fun(fun(A,real),fun(A,$o)),G),H)),Net)
       => ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,C3),Net)
         => ( filterlim(A,real,H,topolo7230453075368039082e_nhds(real,C3),Net)
           => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,C3),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_5225_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Xa: A] :
          ~ ! [A7: fun(nat,set(A))] :
              ( ! [I3: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A7,I3))
             => ( ! [I3: nat] : member(A,Xa,aa(nat,set(A),A7,I3))
               => ~ ! [S9: set(A)] :
                      ( topolo1002775350975398744n_open(A,S9)
                     => ( member(A,Xa,S9)
                       => aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(set(A),fun(nat,$o),aTP_Lamp_tr(fun(nat,set(A)),fun(set(A),fun(nat,$o)),A7),S9)),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_5226_eventually__Inf__base,axiom,
    ! [A: $tType,B2: set(filter(A)),P: fun(A,$o)] :
      ( ( B2 != bot_bot(set(filter(A))) )
     => ( ! [F4: filter(A)] :
            ( member(filter(A),F4,B2)
           => ! [G3: filter(A)] :
                ( member(filter(A),G3,B2)
               => ? [X2: filter(A)] :
                    ( member(filter(A),X2,B2)
                    & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),X2),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F4),G3)) ) ) )
       => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B2))
        <=> ? [X: filter(A)] :
              ( member(filter(A),X,B2)
              & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),X) ) ) ) ) ).

% eventually_Inf_base
tff(fact_5227_eventually__INF__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,$o),F3: fun(A,filter(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F3),A4)))
      <=> ? [Q7: fun(A,fun(B,$o))] :
            ( ! [X: A] :
                ( member(A,X,A4)
               => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),Q7,X)),aa(A,filter(B),F3,X)) )
            & ! [Y4: B] :
                ( ! [X: A] :
                    ( member(A,X,A4)
                   => aa(B,$o,aa(A,fun(B,$o),Q7,X),Y4) )
               => aa(B,$o,P,Y4) ) ) ) ) ).

% eventually_INF_finite
tff(fact_5228_eventually__at__left__real,axiom,
    ! [B3: real,A3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),B3),A3)
     => aa(filter(real),$o,aa(fun(real,$o),fun(filter(real),$o),eventually(real),aa(real,fun(real,$o),aTP_Lamp_ts(real,fun(real,fun(real,$o)),B3),A3)),topolo174197925503356063within(real,A3,aa(real,set(real),set_ord_lessThan(real),A3))) ) ).

% eventually_at_left_real
tff(fact_5229_Bseq__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,F2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( bfun(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_cc(A,fun(fun(nat,A),fun(nat,A)),C3),F2),at_top(nat))
          <=> bfun(nat,A,F2,at_top(nat)) ) ) ) ).

% Bseq_cmult_iff
tff(fact_5230_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xa: A,S3: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xa,S3))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,Xa,S3))
           => ( ( aa(A,B,G,Xa) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ob(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G),topolo174197925503356063within(A,Xa,S3)) ) ) ) ) ).

% differentiable_divide
tff(fact_5231_differentiable__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xa: A,S3: set(A)] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xa,S3))
         => ( ( aa(A,B,F2,Xa) != zero_zero(B) )
           => differentiable(A,B,aTP_Lamp_tt(fun(A,B),fun(A,B),F2),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% differentiable_inverse
tff(fact_5232_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A,S: set(A)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,A3,S))
        <=> ? [D5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
              & ! [X: A] :
                  ( member(A,X,S)
                 => ( ( ( X != A3 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A3)),D5) )
                   => aa(A,$o,P,X) ) ) ) ) ) ).

% eventually_at
tff(fact_5233_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo7230453075368039082e_nhds(A,A3))
        <=> ? [D5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
              & ! [X: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A3)),D5)
                 => aa(A,$o,P,X) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_5234_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( ! [X3: A] :
              ( member(A,X3,set_or5935395276787703475ssThan(A,A3,B3))
             => aa(A,$o,P,X3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,B3,aa(A,set(A),set_ord_lessThan(A),B3))) ) ) ) ).

% eventually_at_leftI
tff(fact_5235_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),A3: A] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),aTP_Lamp_tu(fun(A,$o),fun(A,fun(A,$o)),P),A3)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_5236_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_tv(fun(A,B),fun(B,fun(A,$o)),F2),L)),F3)
         => ( ! [X3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),X3),L)
               => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_th(fun(A,B),fun(B,fun(A,$o)),F2),X3)),F3) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% increasing_tendsto
tff(fact_5237_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,F2: fun(A,B),F3: filter(A)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_tw(B,fun(fun(A,B),fun(A,$o)),L),F2)),F3)
         => ( ! [X3: B] :
                ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),L),X3)
               => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ti(fun(A,B),fun(B,fun(A,$o)),F2),X3)),F3) )
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ) ).

% decreasing_tendsto
tff(fact_5238_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C3: B] :
          ( filterlim(A,B,F2,at_top(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),C3),Z7)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_tx(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),F3) ) ) ) ).

% filterlim_at_top_gt
tff(fact_5239_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F2: fun(A,B),F3: filter(A),C3: B] :
          ( filterlim(A,B,F2,at_bot(B),F3)
        <=> ! [Z7: B] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Z7),C3)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ty(fun(A,B),fun(B,fun(A,$o)),F2),Z7)),F3) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_5240_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F3: filter(A),F2: fun(A,B),Xa: B,G: fun(A,B),Ya: B] :
          ( ( F3 != bot_bot(filter(A)) )
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xa),F3)
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Ya),F3)
             => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G)),F3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Ya),Xa) ) ) ) ) ) ).

% tendsto_le
tff(fact_5241_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),Xa: B,F3: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xa),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,$o)),F2),A3)),F3)
           => ( ( F3 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),Xa) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_5242_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F2: fun(A,B),Xa: B,F3: filter(A),A3: B] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,Xa),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ub(fun(A,B),fun(B,fun(A,$o)),F2),A3)),F3)
           => ( ( F3 != bot_bot(filter(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Xa),A3) ) ) ) ) ).

% tendsto_upperbound
tff(fact_5243_metric__tendsto__imp__tendsto,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(C)
        & real_V7819770556892013058_space(B) )
     => ! [F2: fun(A,B),A3: B,F3: filter(A),G: fun(A,C),B3: C] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_uc(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),F2),A3),G),B3)),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B3),F3) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_5244_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ud(fun(A,real),fun(A,$o),F2)),F3)
       => filterlim(A,real,F2,at_top(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_5245_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,at_infinity(real),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ue(fun(A,real),fun(A,$o),F2)),F3)
       => filterlim(A,real,F2,at_bot(real),F3) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_5246_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: fun(B,filter(A)),B2: set(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F3),B2)))
    <=> ? [X8: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X8),B2)
          & aa(set(B),$o,finite_finite2(B),X8)
          & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),F3),X8))) ) ) ).

% eventually_INF
tff(fact_5247_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F3: filter(A),F2: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F3,F2)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_uf(fun(A,real),fun(A,$o),F2)),F3)
           => topolo3448309680560233919inuous(A,real,F3,aTP_Lamp_sc(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_5248_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A,S: set(A)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,A3,S))
        <=> ? [D5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
              & ! [X: A] :
                  ( member(A,X,S)
                 => ( ( ( X != A3 )
                      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(A,X,A3)),D5) )
                   => aa(A,$o,P,X) ) ) ) ) ) ).

% eventually_at_le
tff(fact_5249_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P3: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P3),at_infinity(A))
        <=> ? [B7: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B7)
              & ! [X: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B7),real_V7770717601297561774m_norm(A,X))
                 => aa(A,$o,P3,X) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_5250_Bseq__eq__bounded,axiom,
    ! [F2: fun(nat,real),A3: real,B3: real] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),aa(set(nat),set(real),image2(nat,real,F2),top_top(set(nat)))),set_or1337092689740270186AtMost(real,A3,B3))
     => bfun(nat,real,F2,at_top(nat)) ) ).

% Bseq_eq_bounded
tff(fact_5251_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L4: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_ti(fun(A,B),fun(B,fun(A,$o)),F2),L4)),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L4,aa(B,set(B),set_ord_lessThan(B),L4)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_5252_tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A),E2: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E2)
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ug(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E2)),F3) ) ) ) ).

% tendstoD
tff(fact_5253_tendstoI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( ! [E: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ug(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E)),F3) )
         => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3) ) ) ).

% tendstoI
tff(fact_5254_tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ug(fun(A,B),fun(B,fun(real,fun(A,$o))),F2),L),E3)),F3) ) ) ) ).

% tendsto_iff
tff(fact_5255_eventually__Inf,axiom,
    ! [A: $tType,P: fun(A,$o),B2: set(filter(A))] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B2))
    <=> ? [X8: set(filter(A))] :
          ( aa(set(filter(A)),$o,aa(set(filter(A)),fun(set(filter(A)),$o),ord_less_eq(set(filter(A))),X8),B2)
          & aa(set(filter(A)),$o,finite_finite2(filter(A)),X8)
          & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X8)) ) ) ).

% eventually_Inf
tff(fact_5256_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(fun(nat,real),fun(nat,$o),aTP_Lamp_uh(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G)),at_top(nat))
         => ( summable(real,G)
           => summable(A,F2) ) ) ) ).

% summable_comparison_test_ev
tff(fact_5257_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [K5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K5)
              & ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N))),K5) ) ) ) ).

% Bseq_def
tff(fact_5258_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K4: real,X4: fun(nat,A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K4)
         => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N2))),K4)
           => bfun(nat,A,X4,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_5259_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
         => ~ ! [K8: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
               => ~ ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N3))),K8) ) ) ) ).

% BseqE
tff(fact_5260_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [N3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N3))),K8) ) ) ) ).

% BseqD
tff(fact_5261_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [N6: nat] :
            ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% Bseq_iff1a
tff(fact_5262_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [N6: nat] :
            ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X4,N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6))) ) ) ).

% Bseq_iff
tff(fact_5263_Bseq__realpow,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => bfun(nat,real,power_power(real,Xa),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_5264_tendsto__arcosh__strong,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),A3)
       => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ui(fun(A,real),fun(A,$o),F2)),F3)
         => filterlim(A,real,aTP_Lamp_pe(fun(A,real),fun(A,real),F2),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F3) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_5265_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A3: A] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P,X3)
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P,X3)
                 => aa(A,$o,Q,aa(B,A,G,X3)) )
             => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_lessThan(A),A3)))
               => ( ! [B5: A] :
                      ( aa(A,$o,Q,B5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),A3) )
                 => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),at_top(B))
                   => filterlim(A,B,F2,at_top(B),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_lessThan(A),A3))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_5266_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B2: set(A),F3: fun(A,filter(B)),P: fun(B,$o)] :
      ( ( B2 != bot_bot(set(A)) )
     => ( ! [A5: A] :
            ( member(A,A5,B2)
           => ! [B5: A] :
                ( member(A,B5,B2)
               => ? [X2: A] :
                    ( member(A,X2,B2)
                    & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(A,filter(B),F3,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F3,A5)),aa(A,filter(B),F3,B5))) ) ) )
       => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),F3),B2)))
        <=> ? [X: A] :
              ( member(A,X,B2)
              & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(A,filter(B),F3,X)) ) ) ) ) ).

% eventually_INF_base
tff(fact_5267_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,C),K4: real] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_uj(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K4)),F3)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ).

% tendsto_0_le
tff(fact_5268_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),C3: B,F3: filter(A),A4: set(B)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,C3),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_uk(fun(A,B),fun(B,fun(set(B),fun(A,$o))),F2),C3),A4)),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,C3,A4),F3) ) ) ) ).

% filterlim_at_withinI
tff(fact_5269_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C3: real,F2: fun(A,B),F3: filter(A)] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C3)
         => ( filterlim(A,B,F2,at_infinity(B),F3)
          <=> ! [R5: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),C3),R5)
               => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aTP_Lamp_ul(fun(A,B),fun(real,fun(A,$o)),F2),R5)),F3) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_5270_tendsto__zero__powrI,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_um(fun(A,real),fun(A,$o),F2)),F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_un(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F3) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_5271_tendsto__powr2,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_um(fun(A,real),fun(A,$o),F2)),F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_un(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B3)),F3) ) ) ) ) ).

% tendsto_powr2
tff(fact_5272_tendsto__powr_H,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F3)
       => ( ( ( A3 != zero_zero(real) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),B3)
              & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_um(fun(A,real),fun(A,$o),F2)),F3) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_un(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B3)),F3) ) ) ) ).

% tendsto_powr'
tff(fact_5273_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_uo(fun(A,B),fun(B,fun(A,$o)),F2),L)),F3) ) ) ) ).

% eventually_floor_less
tff(fact_5274_LIM__at__top__divide,axiom,
    ! [A: $tType,F2: fun(A,real),A3: real,F3: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,A3),F3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),A3)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ud(fun(A,real),fun(A,$o),G)),F3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_up(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G),at_top(real),F3) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_5275_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F2: fun(A,B),L: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L),F3)
         => ( ~ member(B,L,ring_1_Ints(B))
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_uq(fun(A,B),fun(B,fun(A,$o)),F2),L)),F3) ) ) ) ).

% eventually_less_ceiling
tff(fact_5276_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ud(fun(A,real),fun(A,$o),F2)),F3)
       => filterlim(A,real,aTP_Lamp_ur(fun(A,real),fun(A,real),F2),at_top(real),F3) ) ) ).

% filterlim_inverse_at_top
tff(fact_5277_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ud(fun(A,real),fun(A,$o),F2)),F3)
     => ( filterlim(A,real,aTP_Lamp_ur(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_5278_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( filterlim(A,real,F2,topolo7230453075368039082e_nhds(real,zero_zero(real)),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ue(fun(A,real),fun(A,$o),F2)),F3)
       => filterlim(A,real,aTP_Lamp_ur(fun(A,real),fun(A,real),F2),at_bot(real),F3) ) ) ).

% filterlim_inverse_at_bot
tff(fact_5279_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [N6: nat] :
                ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X4,N)),aa(A,A,uminus_uminus(A),aa(nat,A,X4,N6))))),K3) ) ) ) ).

% Bseq_iff3
tff(fact_5280_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X4: fun(nat,A)] :
          ( bfun(nat,A,X4,at_top(nat))
        <=> ? [K3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K3)
              & ? [X: A] :
                ! [N: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X4,N)),aa(A,A,uminus_uminus(A),X)))),K3) ) ) ) ).

% Bseq_iff2
tff(fact_5281_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(fun(nat,real),fun(nat,$o),aTP_Lamp_us(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G)),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_Cauchy'
tff(fact_5282_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F3: filter(A)] :
          ( topolo6773858410816713723filter(A,F3)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [P5: fun(A,$o)] :
                  ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P5),F3)
                  & ! [X: A,Y4: A] :
                      ( ( aa(A,$o,P5,X)
                        & aa(A,$o,P5,Y4) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y4)),E3) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_5283_MVT,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3)
               => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ? [L6: real,Z2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z2)
              & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B3)
              & has_field_derivative(real,F2,L6,topolo174197925503356063within(real,Z2,top_top(set(real))))
              & ( aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B3),A3)),L6) ) ) ) ) ) ).

% MVT
tff(fact_5284_eventually__all__finite,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(A)
     => ! [P: fun(B,fun(A,$o)),Net: filter(B)] :
          ( ! [Y: A] : aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_ut(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y)),Net)
         => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aTP_Lamp_uu(fun(B,fun(A,$o)),fun(B,$o),P)),Net) ) ) ).

% eventually_all_finite
tff(fact_5285_continuous__on__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),F2: fun(A,B),Ta: set(A)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S3)
           => topolo81223032696312382ous_on(A,B,Ta,F2) ) ) ) ).

% continuous_on_subset
tff(fact_5286_IVT2_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),B3: B,Ya: A,A3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,B3)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),aa(B,A,F2,A3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B3)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A3,B3),F2)
               => ? [X3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X3)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B3)
                    & ( aa(B,A,F2,X3) = Ya ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_5287_IVT_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [F2: fun(B,A),A3: B,Ya: A,B3: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,A3)),Ya)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),aa(B,A,F2,B3))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),B3)
             => ( topolo81223032696312382ous_on(B,A,set_or1337092689740270186AtMost(B,A3,B3),F2)
               => ? [X3: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),A3),X3)
                    & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),B3)
                    & ( aa(B,A,F2,X3) = Ya ) ) ) ) ) ) ) ).

% IVT'
tff(fact_5288_continuous__on__empty,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B)] : topolo81223032696312382ous_on(A,B,bot_bot(set(A)),F2) ) ).

% continuous_on_empty
tff(fact_5289_continuous__on__sing,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xa: A,F2: fun(A,B)] : topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))),F2) ) ).

% continuous_on_sing
tff(fact_5290_continuous__on__open__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),Ta: set(A),F2: fun(A,B)] :
          ( topolo1002775350975398744n_open(A,S3)
         => ( topolo1002775350975398744n_open(A,Ta)
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ( topolo81223032696312382ous_on(A,B,Ta,F2)
               => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),Ta),F2) ) ) ) ) ) ).

% continuous_on_open_Un
tff(fact_5291_continuous__image__closed__interval,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ? [C5: real,D6: real] :
            ( ( aa(set(real),set(real),image2(real,real,F2),set_or1337092689740270186AtMost(real,A3,B3)) = set_or1337092689740270186AtMost(real,C5,D6) )
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C5),D6) ) ) ) ).

% continuous_image_closed_interval
tff(fact_5292_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S3: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo81223032696312382ous_on(A,B,S3,G)
           => ( ! [X3: A] :
                  ( member(A,X3,S3)
                 => ( aa(A,B,G,X3) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S3,aa(fun(A,B),fun(A,B),aTP_Lamp_uv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_5293_continuous__on__compose2,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Ta: set(A),G: fun(A,B),S3: set(C),F2: fun(C,A)] :
          ( topolo81223032696312382ous_on(A,B,Ta,G)
         => ( topolo81223032696312382ous_on(C,A,S3,F2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),S3)),Ta)
             => topolo81223032696312382ous_on(C,B,S3,aa(fun(C,A),fun(C,B),aTP_Lamp_uw(fun(A,B),fun(fun(C,A),fun(C,B)),G),F2)) ) ) ) ) ).

% continuous_on_compose2
tff(fact_5294_continuous__on__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S3)
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_ux(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_inverse
tff(fact_5295_continuous__on__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S3)
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S3,aTP_Lamp_uy(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_sgn
tff(fact_5296_continuous__onI__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & dense_order(A)
        & topolo1944317154257567458pology(A) )
     => ! [F2: fun(B,A),A4: set(B)] :
          ( topolo1002775350975398744n_open(A,aa(set(B),set(A),image2(B,A,F2),A4))
         => ( ! [X3: B,Y: B] :
                ( member(B,X3,A4)
               => ( member(B,Y,A4)
                 => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X3),Y)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,X3)),aa(B,A,F2,Y)) ) ) )
           => topolo81223032696312382ous_on(B,A,A4,F2) ) ) ) ).

% continuous_onI_mono
tff(fact_5297_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_uz(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% open_Collect_less
tff(fact_5298_continuous__on__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S3,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S3)
               => ( cos(A,aa(A,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S3,aTP_Lamp_pf(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_tan
tff(fact_5299_open__Collect__less__Int,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,real,S3,G)
           => ? [A7: set(A)] :
                ( topolo1002775350975398744n_open(A,A7)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),S3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_va(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),S3),F2),G)) ) ) ) ) ) ).

% open_Collect_less_Int
tff(fact_5300_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),at_top(A))
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_vb(fun(A,$o),fun(A,$o),P)),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_5301_continuous__on__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S3: set(A),F2: fun(A,A)] :
          ( topolo81223032696312382ous_on(A,A,S3,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S3)
               => ( sin(A,aa(A,A,F2,X3)) != zero_zero(A) ) )
           => topolo81223032696312382ous_on(A,A,S3,aTP_Lamp_pw(fun(A,A),fun(A,A),F2)) ) ) ) ).

% continuous_on_cot
tff(fact_5302_continuous__on__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A4: set(A),F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,A4,F2)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => ( cosh(B,aa(A,B,F2,X3)) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,A4,aTP_Lamp_vc(fun(A,B),fun(A,B),F2)) ) ) ) ).

% continuous_on_tanh
tff(fact_5303_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B),D2: B] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => aa(set(fun(A,B)),$o,finite_finite2(fun(A,B)),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_vd(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A4),B2),D2))) ) ) ).

% finite_set_of_finite_funs
tff(fact_5304_continuous__on__arcosh_H,axiom,
    ! [A4: set(real),F2: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A4,F2)
     => ( ! [X3: real] :
            ( member(real,X3,A4)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,F2,X3)) )
       => topolo81223032696312382ous_on(real,real,A4,aTP_Lamp_ve(fun(real,real),fun(real,real),F2)) ) ) ).

% continuous_on_arcosh'
tff(fact_5305_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_vf(fun(A,$o),fun(A,$o),P)) ) ).

% Least_def
tff(fact_5306_open__Collect__positive,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ? [A7: set(A)] :
              ( topolo1002775350975398744n_open(A,A7)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),S3) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,real),fun(A,$o),aTP_Lamp_vg(set(A),fun(fun(A,real),fun(A,$o)),S3),F2)) ) ) ) ) ).

% open_Collect_positive
tff(fact_5307_continuous__on__powr_H,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,real,S3,G)
           => ( ! [X3: A] :
                  ( member(A,X3,S3)
                 => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F2,X3))
                    & ( ( aa(A,real,F2,X3) = zero_zero(real) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X3)) ) ) )
             => topolo81223032696312382ous_on(A,real,S3,aa(fun(A,real),fun(A,real),aTP_Lamp_vh(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_5308_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( topolo81223032696312382ous_on(A,real,S3,G)
           => ( ! [X3: A] :
                  ( member(A,X3,S3)
                 => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,F2,X3)) )
             => ( ! [X3: A] :
                    ( member(A,X3,S3)
                   => ( aa(A,real,F2,X3) != one_one(real) ) )
               => ( ! [X3: A] :
                      ( member(A,X3,S3)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,G,X3)) )
                 => topolo81223032696312382ous_on(A,real,S3,aa(fun(A,real),fun(A,real),aTP_Lamp_vi(fun(A,real),fun(fun(A,real),fun(A,real)),F2),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_5309_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X3))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X3)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_vj(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arccos
tff(fact_5310_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),F2: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S3,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F2,X3))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F2,X3)),one_one(real)) ) )
           => topolo81223032696312382ous_on(A,real,S3,aTP_Lamp_vk(fun(A,real),fun(A,real),F2)) ) ) ) ).

% continuous_on_arcsin
tff(fact_5311_continuous__on__artanh,axiom,
    ! [A4: set(real)] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),A4),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
     => topolo81223032696312382ous_on(real,real,A4,artanh(real)) ) ).

% continuous_on_artanh
tff(fact_5312_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,B3: A,F2: fun(A,A)] :
          ( ! [X3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X3)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B3)
               => ? [Y2: A] : has_field_derivative(A,F2,Y2,topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A3,B3),F2) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_5313_Rolle__deriv,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),F9: fun(real,fun(real,real))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ( aa(real,real,F2,A3) = aa(real,real,F2,B3) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
         => ( ! [X3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3)
                 => has_derivative(real,real,F2,aa(real,fun(real,real),F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B3)
                & ! [X2: real] : aa(real,real,aa(real,fun(real,real),F9,Z2),X2) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_5314_mvt,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),F9: fun(real,fun(real,real))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3)
               => has_derivative(real,real,F2,aa(real,fun(real,real),F9,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Xi)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xi),B3)
                 => ( aa(real,real,minus_minus(real,aa(real,real,F2,B3)),aa(real,real,F2,A3)) != aa(real,real,aa(real,fun(real,real),F9,Xi),aa(real,real,minus_minus(real,B3),A3)) ) ) ) ) ) ) ).

% mvt
tff(fact_5315_nhds__imp__cauchy__filter,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [F3: filter(A),Xa: A] :
          ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),topolo7230453075368039082e_nhds(A,Xa))
         => topolo6773858410816713723filter(A,F3) ) ) ).

% nhds_imp_cauchy_filter
tff(fact_5316_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B3: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B3),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B3)),topolo174197925503356063within(A,B3,aa(A,set(A),set_ord_lessThan(A),B3))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_5317_DERIV__isconst__end,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( aa(real,real,F2,B3) = aa(real,real,F2,A3) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_5318_DERIV__neg__imp__decreasing__open,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Y2),zero_zero(real)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,B3)),aa(real,real,F2,A3)) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_5319_DERIV__pos__imp__increasing__open,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ! [X3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3)
             => ? [Y2: real] :
                  ( has_field_derivative(real,F2,Y2,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),Y2) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,F2,A3)),aa(real,real,F2,B3)) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_5320_DERIV__isconst2,axiom,
    ! [A3: real,B3: real,F2: fun(real,real),Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
       => ( ! [X3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3)
               => has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A3),Xa)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),B3)
             => ( aa(real,real,F2,Xa) = aa(real,real,F2,A3) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_5321_Rolle,axiom,
    ! [A3: real,B3: real,F2: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => ( ( aa(real,real,F2,A3) = aa(real,real,F2,B3) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A3,B3),F2)
         => ( ! [X3: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),X3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),B3)
                 => differentiable(real,real,F2,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),Z2)
                & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Z2),B3)
                & has_field_derivative(real,F2,zero_zero(real),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_5322_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F2: fun(nat,A),G: fun(nat,real)] :
          ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(fun(nat,real),fun(nat,$o),aTP_Lamp_vl(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),F2),G)),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F2) ) ) ) ).

% summable_bounded_partials
tff(fact_5323_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_vm(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_5324_ord_OLeast__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),P: fun(A,$o)] : aa(fun(A,$o),A,least(A,Less_eq),P) = the(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_vn(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Less_eq),P)) ).

% ord.Least_def
tff(fact_5325_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),$o,finite_finite2(nat),set_or3652927894154168847AtMost(nat,L,U)) ).

% finite_greaterThanAtMost
tff(fact_5326_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_5327_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_5328_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_5329_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_5330_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3) ) ) ).

% infinite_Ioc_iff
tff(fact_5331_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Xa,Ya)) = Ya ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_5332_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Ya,Xa)) = Xa ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_5333_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Xa,Ya)) = Xa ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_5334_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Ya,Xa)) = Ya ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_5335_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,A3,B3) = set_or3652927894154168847AtMost(A,C3,D2) )
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C3) )
            | ( ( A3 = C3 )
              & ( B3 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_5336_ord_OLeast_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : least(A,Less_eq) = least(A,Less_eq) ).

% ord.Least.cong
tff(fact_5337_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B3)),set_or3652927894154168847AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_5338_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ~ aa(set(A),$o,finite_finite2(A),set_or3652927894154168847AtMost(A,A3,B3)) ) ) ).

% infinite_Ioc
tff(fact_5339_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_5340_ivl__disj__int__two_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(6)
tff(fact_5341_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B3: nat] :
      ( ? [X_12: nat] : aa(nat,$o,P,X_12)
     => ( ! [Y: nat] :
            ( aa(nat,$o,P,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),B3) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_5342_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y: nat] :
            ( aa(nat,$o,P,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),B3) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_5343_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y: nat] :
            ( aa(nat,$o,P,Y)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),B3) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_5344_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A3,B3)),set_or3652927894154168847AtMost(A,C3,D2)) = bot_bot(set(A)) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C3)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),A3) ) ) ) ).

% Ioc_disjoint
tff(fact_5345_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_5346_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),Xa: A,Ya: A] :
          ( topolo1002775350975398744n_open(A,S)
         => ( member(A,Xa,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
             => ? [B5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B5),Xa)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B5,Xa)),S) ) ) ) ) ) ).

% open_left
tff(fact_5347_ivl__disj__int__two_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(8)
tff(fact_5348_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_5349_ivl__disj__int__one_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(3)
tff(fact_5350_ivl__disj__int__two_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(2)
tff(fact_5351_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xa: A] :
          ( aa(A,$o,P,Xa)
         => ( ! [Y: A] :
                ( aa(A,$o,P,Y)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) )
           => ( order_Greatest(A,P) = Xa ) ) ) ) ).

% Greatest_equality
tff(fact_5352_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xa: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xa)
         => ( ! [Y: A] :
                ( aa(A,$o,P,Y)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) )
           => ( ! [X3: A] :
                  ( aa(A,$o,P,X3)
                 => ( ! [Y2: A] :
                        ( aa(A,$o,P,Y2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),X3) )
                   => aa(A,$o,Q,X3) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_5353_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = 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,Na))) ) ) ) ).

% sum.head
tff(fact_5354_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,M,Na))) ) ) ) ).

% prod.head
tff(fact_5355_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_5356_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A3,B3)),set_or7035219750837199246ssThan(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),D2) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_5357_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_5358_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C3: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),set_or3652927894154168847AtMost(A,C3,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C3),A3)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_5359_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] : set_or3652927894154168847AtMost(A,A3,B3) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A3,B3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_5360_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_5361_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_5362_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_5363_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),M)
         => ( aa(A,$o,aa(A,fun(A,$o),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_5364_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_5365_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [Na: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(nat,nat,minus_minus(nat,J),I))
     => ( aa(nat,nat,nth(nat,linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J))),Na) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),Na)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5366_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: A,B3: A] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3)))
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,B3)),topolo174197925503356063within(A,B3,aa(A,set(A),set_ord_lessThan(A),B3)))
           => ( ! [X3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),B3)
                   => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,X3)),topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B3),F2) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_5367_eventually__filtercomap__at__topological,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [P: fun(A,$o),F2: fun(A,B),A4: B,B2: set(B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,topolo174197925503356063within(B,A4,B2)))
        <=> ? [S10: set(B)] :
              ( topolo1002775350975398744n_open(B,S10)
              & member(B,A4,S10)
              & ! [X: A] :
                  ( member(B,aa(A,B,F2,X),aa(set(B),set(B),minus_minus(set(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S10),B2)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A4),bot_bot(set(B)))))
                 => aa(A,$o,P,X) ) ) ) ) ).

% eventually_filtercomap_at_topological
tff(fact_5368_at__within__eq,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A,S3: set(A)] : topolo174197925503356063within(A,Xa,S3) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(set(A)),set(filter(A)),image2(set(A),filter(A),aa(set(A),fun(set(A),filter(A)),aTP_Lamp_vo(A,fun(set(A),fun(set(A),filter(A))),Xa),S3)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_vp(A,fun(set(A),$o),Xa)))) ) ).

% at_within_eq
tff(fact_5369_principal__inject,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( principal(A,A4) = principal(A,B2) )
    <=> ( A4 = B2 ) ) ).

% principal_inject
tff(fact_5370_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,aa(A,set(A),set_ord_greaterThan(A),K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),K),I) ) ) ).

% greaterThan_iff
tff(fact_5371_filterlim__filtercomap,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F3: filter(B)] : filterlim(A,B,F2,F3,filtercomap(A,B,F2,F3)) ).

% filterlim_filtercomap
tff(fact_5372_filtercomap__bot,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : filtercomap(A,B,F2,bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtercomap_bot
tff(fact_5373_eventually__filtercomapI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F3: filter(A),F2: fun(B,A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_sr(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2)),filtercomap(B,A,F2,F3)) ) ).

% eventually_filtercomapI
tff(fact_5374_filtercomap__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : filtercomap(A,B,F2,top_top(filter(B))) = top_top(filter(A)) ).

% filtercomap_top
tff(fact_5375_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),Xa)),aa(A,set(A),set_ord_greaterThan(A),Ya))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ).

% greaterThan_subset_iff
tff(fact_5376_principal__le__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),principal(A,A4)),principal(A,B2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ).

% principal_le_iff
tff(fact_5377_inf__principal,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),principal(A,A4)),principal(A,B2)) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) ).

% inf_principal
tff(fact_5378_sup__principal,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),principal(A,A4)),principal(A,B2)) = principal(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ).

% sup_principal
tff(fact_5379_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),top_top(A))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),Xa)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_5380_SUP__principal,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),I5: set(B)] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aTP_Lamp_vq(fun(B,set(A)),fun(B,filter(A)),A4)),I5)) = principal(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5))) ).

% SUP_principal
tff(fact_5381_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_5382_filtercomap__filtercomap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),G: fun(B,C),F3: filter(C)] : filtercomap(A,B,F2,filtercomap(B,C,G,F3)) = filtercomap(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vr(fun(A,B),fun(fun(B,C),fun(A,C)),F2),G),F3) ).

% filtercomap_filtercomap
tff(fact_5383_filtercomap__ident,axiom,
    ! [A: $tType,F3: filter(A)] : filtercomap(A,A,aTP_Lamp_jq(A,A),F3) = F3 ).

% filtercomap_ident
tff(fact_5384_filtercomap__inf,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F1: filter(B),F22: filter(B)] : filtercomap(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F1),F22)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),filtercomap(A,B,F2,F1)),filtercomap(A,B,F2,F22)) ).

% filtercomap_inf
tff(fact_5385_infinite__Ioi,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A3: A] : ~ aa(set(A),$o,finite_finite2(A),aa(A,set(A),set_ord_greaterThan(A),A3)) ) ).

% infinite_Ioi
tff(fact_5386_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xa: A] : aa(A,set(A),set_ord_greaterThan(A),Xa) != bot_bot(set(A)) ) ).

% greaterThan_non_empty
tff(fact_5387_filtercomap__mono,axiom,
    ! [B: $tType,A: $tType,F3: filter(A),F8: filter(A),F2: fun(B,A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F8)
     => aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),filtercomap(B,A,F2,F3)),filtercomap(B,A,F2,F8)) ) ).

% filtercomap_mono
tff(fact_5388_eventually__filtercomap,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F2: fun(A,B),F3: filter(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,F3))
    <=> ? [Q7: fun(B,$o)] :
          ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Q7),F3)
          & ! [X: A] :
              ( aa(B,$o,Q7,aa(A,B,F2,X))
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap
tff(fact_5389_eventually__principal,axiom,
    ! [A: $tType,P: fun(A,$o),S: set(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),principal(A,S))
    <=> ! [X: A] :
          ( member(A,X,S)
         => aa(A,$o,P,X) ) ) ).

% eventually_principal
tff(fact_5390_principal__eq__bot__iff,axiom,
    ! [A: $tType,X4: set(A)] :
      ( ( principal(A,X4) = bot_bot(filter(A)) )
    <=> ( X4 = bot_bot(set(A)) ) ) ).

% principal_eq_bot_iff
tff(fact_5391_bot__eq__principal__empty,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = principal(A,bot_bot(set(A))) ).

% bot_eq_principal_empty
tff(fact_5392_filterlim__iff__le__filtercomap,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F3: filter(B),G2: filter(A)] :
      ( filterlim(A,B,F2,F3,G2)
    <=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),G2),filtercomap(A,B,F2,F3)) ) ).

% filterlim_iff_le_filtercomap
tff(fact_5393_filtercomap__neq__bot,axiom,
    ! [A: $tType,B: $tType,F3: filter(A),F2: fun(B,A)] :
      ( ! [P6: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P6),F3)
         => ? [X2: B] : aa(A,$o,P6,aa(B,A,F2,X2)) )
     => ( filtercomap(B,A,F2,F3) != bot_bot(filter(B)) ) ) ).

% filtercomap_neq_bot
tff(fact_5394_filterlim__principal,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),S: set(B),F3: filter(A)] :
      ( filterlim(A,B,F2,principal(B,S),F3)
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(B),fun(A,$o),aTP_Lamp_vs(fun(A,B),fun(set(B),fun(A,$o)),F2),S)),F3) ) ).

% filterlim_principal
tff(fact_5395_top__eq__principal__UNIV,axiom,
    ! [A: $tType] : top_top(filter(A)) = principal(A,top_top(set(A))) ).

% top_eq_principal_UNIV
tff(fact_5396_nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xa: A] : topolo7230453075368039082e_nhds(A,Xa) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image2(real,filter(A),aTP_Lamp_vu(A,fun(real,filter(A)),Xa)),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% nhds_metric
tff(fact_5397_le__principal,axiom,
    ! [A: $tType,F3: filter(A),A4: set(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),principal(A,A4))
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)),F3) ) ).

% le_principal
tff(fact_5398_INT__greaterThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).

% INT_greaterThan_UNIV
tff(fact_5399_filterlim__If,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G2: filter(B),F3: filter(A),P: fun(A,$o),G: fun(A,B)] :
      ( filterlim(A,B,F2,G2,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),principal(A,aa(fun(A,$o),set(A),collect(A),P))))
     => ( filterlim(A,B,G,G2,aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),principal(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)))))
       => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_vv(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),F2),P),G),G2,F3) ) ) ).

% filterlim_If
tff(fact_5400_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( topolo174197925503356063within(A,Xa,aa(A,set(A),set_ord_greaterThan(A),Xa)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_vw(A,fun(A,filter(A)),Xa)),aa(A,set(A),set_ord_greaterThan(A),Xa))) ) ) ) ).

% at_right_eq
tff(fact_5401_eventually__inf__principal,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),S3: set(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),principal(A,S3)))
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_vx(fun(A,$o),fun(set(A),fun(A,$o)),P),S3)),F3) ) ).

% eventually_inf_principal
tff(fact_5402_filtercomap__sup,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F1: filter(B),F22: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),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,F1)),filtercomap(A,B,F2,F22))),filtercomap(A,B,F2,aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F1),F22))) ).

% filtercomap_sup
tff(fact_5403_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xa: A] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,Xa,aa(A,set(A),set_ord_greaterThan(A),Xa)))
        <=> ? [B7: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B7)
              & ! [Y4: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),B7)
                   => aa(A,$o,P,Y4) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_5404_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xa: A,Ya: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,Xa,aa(A,set(A),set_ord_greaterThan(A),Xa)))
          <=> ? [B7: A] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B7)
                & ! [Y4: A] :
                    ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Y4)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),B7)
                     => aa(A,$o,P,Y4) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_5405_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( topolo174197925503356063within(A,A3,set_or1337092689740270186AtMost(A,A3,B3)) = topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_5406_ivl__disj__int__one_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(7)
tff(fact_5407_filtercomap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),F3: fun(C,filter(B)),B2: set(C)] : filtercomap(A,B,F2,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F3),B2))) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_vy(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F3)),B2)) ).

% filtercomap_INF
tff(fact_5408_nhds__discrete,axiom,
    ! [A: $tType] :
      ( topolo8865339358273720382pology(A)
     => ! [Xa: A] : topolo7230453075368039082e_nhds(A,Xa) = principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ) ).

% nhds_discrete
tff(fact_5409_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_5410_ivl__disj__int__one_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(5)
tff(fact_5411_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xa: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),ord_less(A),Xa)),topolo174197925503356063within(A,Xa,aa(A,set(A),set_ord_greaterThan(A),Xa))) ) ).

% eventually_at_right_less
tff(fact_5412_eventually__filtercomap__at__top__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,at_top(B)))
        <=> ? [N6: B] :
            ! [X: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),N6),aa(A,B,F2,X))
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap_at_top_linorder
tff(fact_5413_eventually__filtercomap__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_top(B) )
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,at_top(B)))
        <=> ? [N6: B] :
            ! [X: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),N6),aa(A,B,F2,X))
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap_at_top_dense
tff(fact_5414_filtercomap__neq__bot__surj,axiom,
    ! [A: $tType,B: $tType,F3: filter(A),F2: fun(B,A)] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
       => ( filtercomap(B,A,F2,F3) != bot_bot(filter(B)) ) ) ) ).

% filtercomap_neq_bot_surj
tff(fact_5415_eventually__filtercomap__at__bot__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N6: B] :
            ! [X: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),N6)
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap_at_bot_linorder
tff(fact_5416_eventually__filtercomap__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(B)
        & no_bot(B) )
     => ! [P: fun(A,$o),F2: fun(A,B)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),filtercomap(A,B,F2,at_bot(B)))
        <=> ? [N6: B] :
            ! [X: A] :
              ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),N6)
             => aa(A,$o,P,X) ) ) ) ).

% eventually_filtercomap_at_bot_dense
tff(fact_5417_filterlim__base,axiom,
    ! [B: $tType,A: $tType,E5: $tType,D: $tType,C: $tType,J4: set(A),I: fun(A,C),I5: set(C),F3: fun(C,set(D)),F2: fun(D,E5),G2: fun(A,set(E5))] :
      ( ! [M4: A,X3: B] :
          ( member(A,M4,J4)
         => member(C,aa(A,C,I,M4),I5) )
     => ( ! [M4: A,X3: D] :
            ( member(A,M4,J4)
           => ( member(D,X3,aa(C,set(D),F3,aa(A,C,I,M4)))
             => member(E5,aa(D,E5,F2,X3),aa(A,set(E5),G2,M4)) ) )
       => filterlim(D,E5,F2,aa(set(filter(E5)),filter(E5),complete_Inf_Inf(filter(E5)),aa(set(A),set(filter(E5)),image2(A,filter(E5),aTP_Lamp_vz(fun(A,set(E5)),fun(A,filter(E5)),G2)),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(C),set(filter(D)),image2(C,filter(D),aTP_Lamp_wa(fun(C,set(D)),fun(C,filter(D)),F3)),I5))) ) ) ).

% filterlim_base
tff(fact_5418_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ? [A5: A,B5: A] :
              ( member(A,Xa,aa(A,set(A),set_ord_lessThan(A),A5))
              & member(A,Ya,aa(A,set(A),set_ord_greaterThan(A),B5))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),A5)),aa(A,set(A),set_ord_greaterThan(A),B5)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_5419_tendsto__principal__singleton,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [F2: fun(A,B),Xa: A] : filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,Xa)),principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))) ) ).

% tendsto_principal_singleton
tff(fact_5420_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( ! [X3: A] :
              ( member(A,X3,set_or5935395276787703475ssThan(A,A3,B3))
             => aa(A,$o,P,X3) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3))) ) ) ) ).

% eventually_at_rightI
tff(fact_5421_nhds__discrete__open,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A] :
          ( topolo1002775350975398744n_open(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))
         => ( topolo7230453075368039082e_nhds(A,Xa) = principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ) ) ) ).

% nhds_discrete_open
tff(fact_5422_greaterThan__0,axiom,
    aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),image2(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_5423_eventually__at__right__real,axiom,
    ! [A3: real,B3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),A3),B3)
     => aa(filter(real),$o,aa(fun(real,$o),fun(filter(real),$o),eventually(real),aa(real,fun(real,$o),aTP_Lamp_ts(real,fun(real,fun(real,$o)),A3),B3)),topolo174197925503356063within(real,A3,aa(real,set(real),set_ord_greaterThan(real),A3))) ) ).

% eventually_at_right_real
tff(fact_5424_filtercomap__SUP,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(A,C),F3: fun(B,filter(C)),B2: set(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(B),set(filter(A)),image2(B,filter(A),aa(fun(B,filter(C)),fun(B,filter(A)),aTP_Lamp_wb(fun(A,C),fun(fun(B,filter(C)),fun(B,filter(A))),F2),F3)),B2))),filtercomap(A,C,F2,aa(set(filter(C)),filter(C),complete_Sup_Sup(filter(C)),aa(set(B),set(filter(C)),image2(B,filter(C),F3),B2)))) ).

% filtercomap_SUP
tff(fact_5425_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_5426_at__bot__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_wc(A,filter(A))),aa(A,set(A),set_ord_atMost(A),C3))) ) ).

% at_bot_sub
tff(fact_5427_filterlim__times__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),P3: B,F1: filter(A),C3: B,L: B] :
          ( filterlim(A,B,F2,topolo174197925503356063within(B,P3,aa(B,set(B),set_ord_greaterThan(B),P3)),F1)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),zero_zero(B)),C3)
           => ( ( L = aa(B,B,aa(B,fun(B,B),times_times(B),C3),P3) )
             => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_wd(fun(A,B),fun(B,fun(A,B)),F2),C3),topolo174197925503356063within(B,L,aa(B,set(B),set_ord_greaterThan(B),L)),F1) ) ) ) ) ).

% filterlim_times_pos
tff(fact_5428_at__within__order,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Xa: A,S3: set(A)] :
          ( ( top_top(set(A)) != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) )
         => ( topolo174197925503356063within(A,Xa,S3) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_we(A,fun(set(A),fun(A,filter(A))),Xa),S3)),aa(A,set(A),set_ord_greaterThan(A),Xa)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aa(set(A),fun(A,filter(A)),aTP_Lamp_wf(A,fun(set(A),fun(A,filter(A))),Xa),S3)),aa(A,set(A),set_ord_lessThan(A),Xa)))) ) ) ) ).

% at_within_order
tff(fact_5429_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F2: fun(A,B),L4: B,F3: filter(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,L4),F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(B,fun(A,$o),aTP_Lamp_th(fun(A,B),fun(B,fun(A,$o)),F2),L4)),F3)
           => filterlim(A,B,F2,topolo174197925503356063within(B,L4,aa(B,set(B),set_ord_greaterThan(B),L4)),F3) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_5430_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F3: fun(A,set(B)),F2: fun(B,C),G2: fun(D,set(C)),J4: set(D)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => ! [J2: A] :
                ( member(A,J2,I5)
               => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F3,I2)),aa(A,set(B),F3,J2))
                  | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F3,J2)),aa(A,set(B),F3,I2)) ) ) )
       => ( filterlim(B,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image2(D,filter(C),aTP_Lamp_wg(fun(D,set(C)),fun(D,filter(C)),G2)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_wh(fun(A,set(B)),fun(A,filter(B)),F3)),I5)))
        <=> ! [X: D] :
              ( member(D,X,J4)
             => ? [Xa3: A] :
                  ( member(A,Xa3,I5)
                  & ! [Xb4: B] :
                      ( member(B,Xb4,aa(A,set(B),F3,Xa3))
                     => member(C,aa(B,C,F2,Xb4),aa(D,set(C),G2,X)) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_5431_INF__principal__finite,axiom,
    ! [B: $tType,A: $tType,X4: set(A),F2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),X4)
     => ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),aTP_Lamp_wh(fun(A,set(B)),fun(A,filter(B)),F2)),X4)) = principal(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),X4))) ) ) ).

% INF_principal_finite
tff(fact_5432_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)),image2(real,filter(A),aTP_Lamp_wj(real,filter(A))),top_top(set(real)))) ) ) ).

% at_infinity_def
tff(fact_5433_at__bot__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_bot(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_wk(A,filter(A))),top_top(set(A)))) ) ) ).

% at_bot_def
tff(fact_5434_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B3: A,F2: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B3),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
           => filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,F2,A3)),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_5435_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),F2: fun(A,B),P: fun(B,$o),G: fun(B,A),A3: A] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,Q,X3)
             => ( aa(A,$o,Q,Y)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) ) ) )
         => ( ! [X3: B] :
                ( aa(B,$o,P,X3)
               => ( aa(A,B,F2,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( aa(B,$o,P,X3)
                 => aa(A,$o,Q,aa(B,A,G,X3)) )
             => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3)))
               => ( ! [B5: A] :
                      ( aa(A,$o,Q,B5)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B5) )
                 => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),at_bot(B))
                   => filterlim(A,B,F2,at_bot(B),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_5436_at__within__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A,S3: set(A)] : topolo174197925503356063within(A,A3,S3) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),topolo7230453075368039082e_nhds(A,A3)),principal(A,aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ).

% at_within_def
tff(fact_5437_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
         => ( topolo174197925503356063within(A,Xa,aa(A,set(A),set_ord_lessThan(A),Xa)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_wl(A,fun(A,filter(A)),Xa)),aa(A,set(A),set_ord_lessThan(A),Xa))) ) ) ) ).

% at_left_eq
tff(fact_5438_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,G: fun(A,B),F2: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_lessThan(A),A3)),G)
         => ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,aa(A,B,G,A3)),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_wm(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),G),F2)) ) ) ) ).

% isCont_If_ge
tff(fact_5439_complete__uniform,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S: set(A)] :
          ( topolo2479028161051973599mplete(A,S)
        <=> ! [F10: filter(A)] :
              ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F10),principal(A,S))
             => ( ( F10 != bot_bot(filter(A)) )
               => ( topolo6773858410816713723filter(A,F10)
                 => ? [X: A] :
                      ( member(A,X,S)
                      & aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F10),topolo7230453075368039082e_nhds(A,X)) ) ) ) ) ) ) ).

% complete_uniform
tff(fact_5440_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A)] :
          ( ! [A5: A,B5: A,X3: A] :
              ( member(A,A5,S)
             => ( member(A,B5,S)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),X3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),B5)
                   => member(A,X3,S) ) ) ) )
         => ? [A5: A,B5: A] :
              ( ( S = bot_bot(set(A)) )
              | ( S = top_top(set(A)) )
              | ( S = aa(A,set(A),set_ord_lessThan(A),B5) )
              | ( S = aa(A,set(A),set_ord_atMost(A),B5) )
              | ( S = aa(A,set(A),set_ord_greaterThan(A),A5) )
              | ( S = set_ord_atLeast(A,A5) )
              | ( S = set_or5935395276787703475ssThan(A,A5,B5) )
              | ( S = set_or3652927894154168847AtMost(A,A5,B5) )
              | ( S = set_or7035219750837199246ssThan(A,A5,B5) )
              | ( S = set_or1337092689740270186AtMost(A,A5,B5) ) ) ) ) ).

% interval_cases
tff(fact_5441_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( ! [F5: fun(nat,A)] :
                ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(nat,A,F5,N3))
               => ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F5,N3)),B3)
                 => ( order_antimono(nat,A,F5)
                   => ( filterlim(nat,A,F5,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(fun(nat,A),fun(nat,$o),aTP_Lamp_wn(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F5)),at_top(nat)) ) ) ) )
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_5442_atLeast__0,axiom,
    set_ord_atLeast(nat,zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_5443_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_atLeast(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),I) ) ) ).

% atLeast_iff
tff(fact_5444_atLeast__empty__triv,axiom,
    ! [A: $tType] : set_ord_atLeast(set(A),bot_bot(set(A))) = top_top(set(set(A))) ).

% atLeast_empty_triv
tff(fact_5445_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xa: A,Ya: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,Xa)),set_ord_atLeast(A,Ya))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa) ) ) ).

% atLeast_subset_iff
tff(fact_5446_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atLeast(A,L2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L2),L) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_5447_not__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A] : bot_bot(set(A)) != set_ord_atLeast(A,L) ) ).

% not_empty_eq_Ici_eq_empty
tff(fact_5448_infinite__Ici,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A3: A] : ~ aa(set(A),$o,finite_finite2(A),set_ord_atLeast(A,A3)) ) ).

% infinite_Ici
tff(fact_5449_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ya)),aa(A,B,F2,Xa)) ) ) ) ).

% antimonoD
tff(fact_5450_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( order_antimono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ya)),aa(A,B,F2,Xa)) ) ) ) ).

% antimonoE
tff(fact_5451_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y)),aa(A,B,F2,X3)) )
         => order_antimono(A,B,F2) ) ) ).

% antimonoI
tff(fact_5452_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_antimono(A,B,F2)
        <=> ! [X: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Y4)),aa(A,B,F2,X)) ) ) ) ).

% antimono_def
tff(fact_5453_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : set_ord_atLeast(A,L) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_5454_atLeast__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] :
          ( ( set_ord_atLeast(A,Xa) = top_top(set(A)) )
        <=> ( Xa = bot_bot(A) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_5455_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_ord_atLeast(A,L)) ) ).

% not_UNIV_le_Ici
tff(fact_5456_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L2: A,H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,L)),set_or1337092689740270186AtMost(A,L2,H2)) ) ).

% not_Ici_le_Icc
tff(fact_5457_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,L)),aa(A,set(A),set_ord_atMost(A),H2)) ) ).

% not_Ici_le_Iic
tff(fact_5458_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L2: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_ord_atLeast(A,L2)) ) ).

% not_Iic_le_Ici
tff(fact_5459_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),A3)),set_ord_atLeast(A,A3)) ) ).

% Ioi_le_Ico
tff(fact_5460_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: fun(nat,A),I: nat] :
          ( order_antimono(nat,A,A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A4,aa(nat,nat,suc,I))),aa(nat,A,A4,I)) ) ) ).

% decseq_SucD
tff(fact_5461_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,aa(nat,nat,suc,N2))),aa(nat,A,X4,N2))
         => order_antimono(nat,A,X4) ) ) ).

% decseq_SucI
tff(fact_5462_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_antimono(nat,A,F2)
        <=> ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,aa(nat,nat,suc,N))),aa(nat,A,F2,N)) ) ) ).

% decseq_Suc_iff
tff(fact_5463_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( order_antimono(nat,A,X4)
        <=> ! [M2: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N)),aa(nat,A,X4,M2)) ) ) ) ).

% decseq_def
tff(fact_5464_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( order_antimono(nat,A,F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,J)),aa(nat,A,F2,I)) ) ) ) ).

% decseqD
tff(fact_5465_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),set_ord_atLeast(A,U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_5466_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atLeast(A,A3)),aa(A,set(A),set_ord_greaterThan(A),B3))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_5467_ivl__disj__int__one_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,U)),set_ord_atLeast(A,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(8)
tff(fact_5468_ivl__disj__int__one_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,U)),set_ord_atLeast(A,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(6)
tff(fact_5469_decseq__bounded,axiom,
    ! [X4: fun(nat,real),B2: real] :
      ( order_antimono(nat,real,X4)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),aa(nat,real,X4,I2))
       => bfun(nat,real,X4,at_top(nat)) ) ) ).

% decseq_bounded
tff(fact_5470_atMost__Int__atLeast,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Na: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),Na)),set_ord_atLeast(A,Na)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Na),bot_bot(set(A))) ) ).

% atMost_Int_atLeast
tff(fact_5471_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = set_ord_atLeast(A,L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_5472_ivl__disj__un__singleton_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = set_ord_atLeast(A,L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_5473_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),L),U)
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),set_ord_atLeast(A,U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_5474_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X4: fun(nat,A),L4: A,Na: nat] :
          ( order_antimono(nat,A,X4)
         => ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L4),aa(nat,A,X4,Na)) ) ) ) ).

% decseq_ge
tff(fact_5475_decseq__convergent,axiom,
    ! [X4: fun(nat,real),B2: real] :
      ( order_antimono(nat,real,X4)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),aa(nat,real,X4,I2))
       => ~ ! [L5: real] :
              ( filterlim(nat,real,X4,topolo7230453075368039082e_nhds(real,L5),at_top(nat))
             => ~ ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),L5),aa(nat,real,X4,I3)) ) ) ) ).

% decseq_convergent
tff(fact_5476_at__top__sub,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C3: A] : at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_wo(A,filter(A))),set_ord_atLeast(A,C3))) ) ).

% at_top_sub
tff(fact_5477_atLeast__Suc,axiom,
    ! [K: nat] : set_ord_atLeast(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_atLeast(nat,K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_5478_continuous__on__arcosh,axiom,
    ! [A4: set(real)] :
      ( aa(set(real),$o,aa(set(real),fun(set(real),$o),ord_less_eq(set(real)),A4),set_ord_atLeast(real,one_one(real)))
     => topolo81223032696312382ous_on(real,real,A4,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_5479_at__top__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( at_top(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image2(A,filter(A),aTP_Lamp_wp(A,filter(A))),top_top(set(A)))) ) ) ).

% at_top_def
tff(fact_5480_tendsto__at__right__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,B3: A,X4: fun(A,B),L4: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( ! [S2: fun(nat,A)] :
                ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(nat,A,S2,N3))
               => ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S2,N3)),B3)
                 => ( order_antimono(nat,A,S2)
                   => ( filterlim(nat,A,S2,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wq(fun(A,B),fun(fun(nat,A),fun(nat,B)),X4),S2),topolo7230453075368039082e_nhds(B,L4),at_top(nat)) ) ) ) )
           => filterlim(A,B,X4,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_greaterThan(A),A3))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_5481_Gcd__eq__Max,axiom,
    ! [M5: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),M5)
     => ( ( M5 != bot_bot(set(nat)) )
       => ( ~ member(nat,zero_zero(nat),M5)
         => ( gcd_Gcd(nat,M5) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image2(nat,set(nat),aTP_Lamp_wr(nat,set(nat))),M5))) ) ) ) ) ).

% Gcd_eq_Max
tff(fact_5482_continuous__at__Sup__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( order_antimono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S),aa(A,set(A),set_ord_lessThan(A),aa(set(A),A,complete_Sup_Sup(A),S))),F2)
           => ( ( S != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S)
               => ( aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),S)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),S)) ) ) ) ) ) ) ).

% continuous_at_Sup_antimono
tff(fact_5483_continuous__at__Inf__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( order_antimono(A,B,F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Inf_Inf(A),S),aa(A,set(A),set_ord_greaterThan(A),aa(set(A),A,complete_Inf_Inf(A),S))),F2)
           => ( ( S != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S)
               => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),S)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),S)) ) ) ) ) ) ) ).

% continuous_at_Inf_antimono
tff(fact_5484_bdd__belowI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A),M: A] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X3) )
         => condit1013018076250108175_below(A,A4) ) ) ).

% bdd_belowI
tff(fact_5485_bdd__below_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A),M5: A] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M5),X3) )
         => condit1013018076250108175_below(A,A4) ) ) ).

% bdd_below.I
tff(fact_5486_bdd__above_OI,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A),M5: A] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),M5) )
         => condit941137186595557371_above(A,A4) ) ) ).

% bdd_above.I
tff(fact_5487_bdd__below__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit1013018076250108175_below(A,bot_bot(set(A))) ) ).

% bdd_below_empty
tff(fact_5488_bdd__above__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => condit941137186595557371_above(A,bot_bot(set(A))) ) ).

% bdd_above_empty
tff(fact_5489_bdd__below__insert,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: A,A4: set(A)] :
          ( condit1013018076250108175_below(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))
        <=> condit1013018076250108175_below(A,A4) ) ) ).

% bdd_below_insert
tff(fact_5490_bdd__above__insert,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: A,A4: set(A)] :
          ( condit941137186595557371_above(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))
        <=> condit941137186595557371_above(A,A4) ) ) ).

% bdd_above_insert
tff(fact_5491_bdd__below__Un,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( condit1013018076250108175_below(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
        <=> ( condit1013018076250108175_below(A,A4)
            & condit1013018076250108175_below(A,B2) ) ) ) ).

% bdd_below_Un
tff(fact_5492_bdd__above__Un,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( condit941137186595557371_above(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
        <=> ( condit941137186595557371_above(A,A4)
            & condit941137186595557371_above(A,B2) ) ) ) ).

% bdd_above_Un
tff(fact_5493_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Max_singleton
tff(fact_5494_bdd__above__image__sup,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [F2: fun(B,A),G: fun(B,A),A4: set(B)] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ws(fun(B,A),fun(fun(B,A),fun(B,A)),F2),G)),A4))
        <=> ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A4))
            & condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,G),A4)) ) ) ) ).

% bdd_above_image_sup
tff(fact_5495_Max__divisors__self__nat,axiom,
    ! [Na: nat] :
      ( ( Na != zero_zero(nat) )
     => ( aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bi(nat,fun(nat,$o),Na))) = Na ) ) ).

% Max_divisors_self_nat
tff(fact_5496_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),Xa)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_5497_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),Xa)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa) ) ) ) ) ) ).

% Max_less_iff
tff(fact_5498_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A4: set(A),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_wt(B,fun(A,B),C3)),A4)) = C3 ) ) ) ) ).

% Max_const
tff(fact_5499_bdd__below__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I5: set(A),A4: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( condit1013018076250108175_below(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5)))
          <=> ! [X: A] :
                ( member(A,X,I5)
               => condit1013018076250108175_below(B,aa(A,set(B),A4,X)) ) ) ) ) ).

% bdd_below_UN
tff(fact_5500_bdd__above__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( lattice(B)
     => ! [I5: set(A),A4: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( condit941137186595557371_above(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5)))
          <=> ! [X: A] :
                ( member(A,X,I5)
               => condit941137186595557371_above(B,aa(A,set(B),A4,X)) ) ) ) ) ).

% bdd_above_UN
tff(fact_5501_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) ) ) ) ) ).

% Max_insert
tff(fact_5502_cInf__le__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A4)
           => ( condit1013018076250108175_below(A,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ) ).

% cInf_le_cSup
tff(fact_5503_bdd__below_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A4: set(A),M5: B,F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M5),aa(A,B,F2,X3)) )
         => condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A4)) ) ) ).

% bdd_below.I2
tff(fact_5504_bdd__belowI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A4: set(A),M: B,F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),M),aa(A,B,F2,X3)) )
         => condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A4)) ) ) ).

% bdd_belowI2
tff(fact_5505_bdd__above_OI2,axiom,
    ! [B: $tType,A: $tType] :
      ( preorder(B)
     => ! [A4: set(A),F2: fun(A,B),M5: B] :
          ( ! [X3: A] :
              ( member(A,X3,A4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),M5) )
         => condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A4)) ) ) ).

% bdd_above.I2
tff(fact_5506_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_5507_bdd__above__nat,axiom,
    ! [X4: set(nat)] :
      ( condit941137186595557371_above(nat,X4)
    <=> aa(set(nat),$o,finite_finite2(nat),X4) ) ).

% bdd_above_nat
tff(fact_5508_bdd__below__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => condit1013018076250108175_below(A,A4) ) ) ).

% bdd_below_finite
tff(fact_5509_bdd__above__finite,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => condit941137186595557371_above(A,A4) ) ) ).

% bdd_above_finite
tff(fact_5510_bdd__below__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: set(A),A4: set(A)] :
          ( condit1013018076250108175_below(A,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => condit1013018076250108175_below(A,A4) ) ) ) ).

% bdd_below_mono
tff(fact_5511_bdd__below_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A)] :
          ( condit1013018076250108175_below(A,A4)
         => ~ ! [M9: A] :
                ~ ! [X2: A] :
                    ( member(A,X2,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M9),X2) ) ) ) ).

% bdd_below.E
tff(fact_5512_bdd__below_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A)] :
          ( condit1013018076250108175_below(A,A4)
        <=> ? [M8: A] :
            ! [X: A] :
              ( member(A,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M8),X) ) ) ) ).

% bdd_below.unfold
tff(fact_5513_bdd__above_OE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A)] :
          ( condit941137186595557371_above(A,A4)
         => ~ ! [M9: A] :
                ~ ! [X2: A] :
                    ( member(A,X2,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),M9) ) ) ) ).

% bdd_above.E
tff(fact_5514_bdd__above_Ounfold,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: set(A)] :
          ( condit941137186595557371_above(A,A4)
        <=> ? [M8: A] :
            ! [X: A] :
              ( member(A,X,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M8) ) ) ) ).

% bdd_above.unfold
tff(fact_5515_bdd__above__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: set(A),A4: set(A)] :
          ( condit941137186595557371_above(A,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => condit941137186595557371_above(A,A4) ) ) ) ).

% bdd_above_mono
tff(fact_5516_cInf__lower,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A,X4: set(A)] :
          ( member(A,Xa,X4)
         => ( condit1013018076250108175_below(A,X4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X4)),Xa) ) ) ) ).

% cInf_lower
tff(fact_5517_cInf__lower2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A,X4: set(A),Ya: A] :
          ( member(A,Xa,X4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => ( condit1013018076250108175_below(A,X4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X4)),Ya) ) ) ) ) ).

% cInf_lower2
tff(fact_5518_cSup__upper2,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A,X4: set(A),Ya: A] :
          ( member(A,Xa,X4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
           => ( condit941137186595557371_above(A,X4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),aa(set(A),A,complete_Sup_Sup(A),X4)) ) ) ) ) ).

% cSup_upper2
tff(fact_5519_cSup__upper,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Xa: A,X4: set(A)] :
          ( member(A,Xa,X4)
         => ( condit941137186595557371_above(A,X4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),X4)) ) ) ) ).

% cSup_upper
tff(fact_5520_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) ) ) ) ).

% Max.coboundedI
tff(fact_5521_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ! [X3: A] :
                  ( member(A,X3,A4)
                 => ? [Xa2: A] :
                      ( member(A,Xa2,B2)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa2) ) )
             => ( ! [X3: A] :
                    ( member(A,X3,B2)
                   => ? [Xa2: A] :
                        ( member(A,Xa2,A4)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa2) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = aa(set(A),A,lattic643756798349783984er_Max(A),B2) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_5522_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [Y: A] :
                ( member(A,Y,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xa) )
           => ( member(A,Xa,A4)
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = Xa ) ) ) ) ) ).

% Max_eqI
tff(fact_5523_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) ) ) ) ).

% Max_ge
tff(fact_5524_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A4),A4) ) ) ) ).

% Max_in
tff(fact_5525_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) = aa(set(A),A,lattic643756798349783984er_Max(A),A4) ) ) ) ) ).

% Max.in_idem
tff(fact_5526_cINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Xa: B,U: A] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A4))
         => ( member(B,Xa,A4)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F2,Xa)),U)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4))),U) ) ) ) ) ).

% cINF_lower2
tff(fact_5527_cINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Xa: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A4))
         => ( member(B,Xa,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4))),aa(B,A,F2,Xa)) ) ) ) ).

% cINF_lower
tff(fact_5528_cInf__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B2: set(A),A4: set(A)] :
          ( ( B2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A4)
           => ( ! [B5: A] :
                  ( member(A,B5,B2)
                 => ? [X2: A] :
                      ( member(A,X2,A4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B5) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ) ) ).

% cInf_mono
tff(fact_5529_le__cInf__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(set(A),A,complete_Inf_Inf(A),S))
            <=> ! [X: A] :
                  ( member(A,X,S)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),X) ) ) ) ) ) ).

% le_cInf_iff
tff(fact_5530_cSUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Xa: A,A4: set(A),F2: fun(A,B)] :
          ( member(A,Xa,A4)
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ).

% cSUP_upper
tff(fact_5531_cSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Xa: B,U: A] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A4))
         => ( member(B,Xa,A4)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(B,A,F2,Xa))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4))) ) ) ) ) ).

% cSUP_upper2
tff(fact_5532_cInf__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Ya: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X4)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X4)),Ya)
            <=> ? [X: A] :
                  ( member(A,X,X4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Ya) ) ) ) ) ) ).

% cInf_less_iff
tff(fact_5533_cSup__le__iff,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A),A3: A] :
          ( ( S != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),S)),A3)
            <=> ! [X: A] :
                  ( member(A,X,S)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),A3) ) ) ) ) ) ).

% cSup_le_iff
tff(fact_5534_cSup__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [B2: set(A),A4: set(A)] :
          ( ( B2 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A4)
           => ( ! [B5: A] :
                  ( member(A,B5,B2)
                 => ? [X2: A] :
                      ( member(A,X2,A4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B5),X2) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),B2)),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ) ).

% cSup_mono
tff(fact_5535_less__cSup__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A),Ya: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X4)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),aa(set(A),A,complete_Sup_Sup(A),X4))
            <=> ? [X: A] :
                  ( member(A,X,X4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),X) ) ) ) ) ) ).

% less_cSup_iff
tff(fact_5536_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),Xa) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),Xa) ) ) ) ) ).

% Max.boundedI
tff(fact_5537_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),Xa)
             => ! [A10: A] :
                  ( member(A,A10,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A10),Xa) ) ) ) ) ) ).

% Max.boundedE
tff(fact_5538_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A4) )
            <=> ( member(A,M,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_5539_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A4))
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_5540_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = M )
            <=> ( member(A,M,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_5541_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A4))
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_5542_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [B5: A] :
                ( member(A,B5,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B5),A3) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = A3 ) ) ) ) ).

% Max_insert2
tff(fact_5543_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = aa(set(A),A,complete_Sup_Sup(A),A4) ) ) ) ) ).

% Max_Sup
tff(fact_5544_cSup__eq__Max,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X4) = aa(set(A),A,lattic643756798349783984er_Max(A),X4) ) ) ) ) ).

% cSup_eq_Max
tff(fact_5545_less__cINF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Ya: A,I: B] :
          ( condit1013018076250108175_below(A,aa(set(B),set(A),image2(B,A,F2),A4))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image2(B,A,F2),A4)))
           => ( member(B,I,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),aa(B,A,F2,I)) ) ) ) ) ).

% less_cINF_D
tff(fact_5546_cSUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [F2: fun(B,A),A4: set(B),Ya: A,I: B] :
          ( condit941137186595557371_above(A,aa(set(B),set(A),image2(B,A,F2),A4))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),A4))),Ya)
           => ( member(B,I,A4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,F2,I)),Ya) ) ) ) ) ).

% cSUP_lessD
tff(fact_5547_le__cINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),U: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4)))
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,X)) ) ) ) ) ) ).

% le_cINF_iff
tff(fact_5548_cINF__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [B2: set(A),F2: fun(C,B),A4: set(C),G: fun(A,B)] :
          ( ( B2 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(C),set(B),image2(C,B,F2),A4))
           => ( ! [M4: A] :
                  ( member(A,M4,B2)
                 => ? [X2: C] :
                      ( member(C,X2,A4)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(C,B,F2,X2)),aa(A,B,G,M4)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,F2),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B2))) ) ) ) ) ).

% cINF_mono
tff(fact_5549_cInf__superset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,B2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),B2)),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ) ) ) ).

% cInf_superset_mono
tff(fact_5550_cSUP__mono,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(C,B),B2: set(C),F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(C),set(B),image2(C,B,G),B2))
           => ( ! [N2: A] :
                  ( member(A,N2,A4)
                 => ? [X2: C] :
                      ( member(C,X2,B2)
                      & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,N2)),aa(C,B,G,X2)) ) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,G),B2))) ) ) ) ) ).

% cSUP_mono
tff(fact_5551_cSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),U: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),U)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),U) ) ) ) ) ) ).

% cSUP_le_iff
tff(fact_5552_cSup__subset__mono,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,B2)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ) ) ).

% cSup_subset_mono
tff(fact_5553_cInf__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),A3: A] :
          ( condit1013018076250108175_below(A,X4)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X4)) = $ite(X4 = bot_bot(set(A)),A3,aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X4))) ) ) ) ).

% cInf_insert_If
tff(fact_5554_cInf__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),A3: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,X4)
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),X4)) ) ) ) ) ).

% cInf_insert
tff(fact_5555_cSup__insert,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),A3: A] :
          ( ( X4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,X4)
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X4)) ) ) ) ) ).

% cSup_insert
tff(fact_5556_cSup__insert__If,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X4: set(A),A3: A] :
          ( condit941137186595557371_above(A,X4)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X4)) = $ite(X4 = bot_bot(set(A)),A3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),X4))) ) ) ) ).

% cSup_insert_If
tff(fact_5557_cInf__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,A4)
           => ( ( B2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,B2)
               => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ) ) ) ) ).

% cInf_union_distrib
tff(fact_5558_cSup__union__distrib,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,A4)
           => ( ( B2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,B2)
               => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ) ) ) ) ).

% cSup_union_distrib
tff(fact_5559_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),aa(set(A),A,lattic643756798349783984er_Max(A),B2)) ) ) ) ) ).

% Max.subset_imp
tff(fact_5560_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M5: set(A),N4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M5),N4)
         => ( ( M5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M5)),aa(set(A),A,lattic643756798349783984er_Max(A),N4)) ) ) ) ) ).

% Max_mono
tff(fact_5561_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N4: set(A)] :
          ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X3)),aa(A,A,H,Y))
         => ( aa(set(A),$o,finite_finite2(A),N4)
           => ( ( N4 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798349783984er_Max(A),N4)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_5562_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( B2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B2)),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) = aa(set(A),A,lattic643756798349783984er_Max(A),A4) ) ) ) ) ) ).

% Max.subset
tff(fact_5563_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),A4)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_5564_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic643756798349783984er_Max(A),A4),A4) ) ) ) ) ).

% Max.closed
tff(fact_5565_cINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A4: set(A),F2: fun(A,B),A3: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),A3)
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),A3) ) ) ) ) ) ).

% cINF_less_iff
tff(fact_5566_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A4)),aa(set(A),A,lattic643756798349783984er_Max(A),B2)) ) ) ) ) ) ) ).

% Max.union
tff(fact_5567_less__cSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( condit6923001295902523014norder(B)
     => ! [A4: set(A),F2: fun(A,B),A3: B] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4)))
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(A,B,F2,X)) ) ) ) ) ) ).

% less_cSUP_iff
tff(fact_5568_cINF__inf__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),A4))
             => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),A4))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wu(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A4)) ) ) ) ) ) ).

% cINF_inf_distrib
tff(fact_5569_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),A4))
             => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),A4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wv(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A4)) ) ) ) ) ) ).

% conditionally_complete_lattice_class.SUP_sup_distrib
tff(fact_5570_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = finite_fold(A,A,ord_max(A),Xa,A4) ) ) ) ).

% Max.eq_fold
tff(fact_5571_card__le__Suc__Max,axiom,
    ! [S: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),S)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S))) ) ).

% card_le_Suc_Max
tff(fact_5572_Sup__nat__def,axiom,
    ! [X4: set(nat)] :
      aa(set(nat),nat,complete_Sup_Sup(nat),X4) = $ite(X4 = bot_bot(set(nat)),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),X4)) ).

% Sup_nat_def
tff(fact_5573_divide__nat__def,axiom,
    ! [M: nat,Na: nat] :
      divide_divide(nat,M,Na) = $ite(Na = zero_zero(nat),zero_zero(nat),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ww(nat,fun(nat,fun(nat,$o)),M),Na)))) ).

% divide_nat_def
tff(fact_5574_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F2: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_wx(fun(A,B),fun(B,fun(A,B)),F2),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F2),S))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_5575_cINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(A,B),B2: set(A),F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,G),B2))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
             => ( ! [X3: A] :
                    ( member(A,X3,B2)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,G,X3)),aa(A,B,F2,X3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B2))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ) ) ).

% cINF_superset_mono
tff(fact_5576_cSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),G: fun(A,B),B2: set(A),F2: fun(A,B)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,G),B2))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
             => ( ! [X3: A] :
                    ( member(A,X3,A4)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B2))) ) ) ) ) ) ).

% cSUP_subset_mono
tff(fact_5577_less__eq__cInf__inter,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( condit1013018076250108175_below(A,A4)
         => ( condit1013018076250108175_below(A,B2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ) ).

% less_eq_cInf_inter
tff(fact_5578_cINF__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),A3: A] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A3)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ) ).

% cINF_insert
tff(fact_5579_cSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( condit941137186595557371_above(A,A4)
         => ( condit941137186595557371_above(A,B2)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) != bot_bot(set(A)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2))) ) ) ) ) ).

% cSup_inter_less_eq
tff(fact_5580_cSUP__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),A3: A] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A3)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ) ).

% cSUP_insert
tff(fact_5581_cINF__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),B2: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( ( B2 != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(B,aa(set(A),set(B),image2(A,B,F2),B2))
               => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),B2))) ) ) ) ) ) ) ).

% cINF_union
tff(fact_5582_cSUP__union,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [A4: set(A),F2: fun(A,B),B2: set(A)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),A4))
           => ( ( B2 != bot_bot(set(A)) )
             => ( condit941137186595557371_above(B,aa(set(A),set(B),image2(A,B,F2),B2))
               => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),B2))) ) ) ) ) ) ) ).

% cSUP_union
tff(fact_5583_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_5584_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_5585_cINF__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(A),B2: fun(A,set(B)),F2: fun(B,C)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => ( aa(A,set(B),B2,X3) != bot_bot(set(B)) ) )
           => ( condit1013018076250108175_below(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_wy(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B2),F2)),A4)))
             => ( aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4)))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_wz(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B2),F2)),A4)) ) ) ) ) ) ).

% cINF_UNION
tff(fact_5586_cSUP__UNION,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [A4: set(A),B2: fun(A,set(B)),F2: fun(B,C)] :
          ( ( A4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => ( aa(A,set(B),B2,X3) != bot_bot(set(B)) ) )
           => ( condit941137186595557371_above(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_wy(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),B2),F2)),A4)))
             => ( aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,F2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4)))) = aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xa(fun(A,set(B)),fun(fun(B,C),fun(A,C)),B2),F2)),A4)) ) ) ) ) ) ).

% cSUP_UNION
tff(fact_5587_sum__le__card__Max,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image2(A,nat,F2),A4)))) ) ).

% sum_le_card_Max
tff(fact_5588_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_ta(A,fun(A,$o))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_5589_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))),image2(real,filter(product_prod(A,A)),aTP_Lamp_xc(real,filter(product_prod(A,A)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ).

% uniformity_dist
tff(fact_5590_compactE__image,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),C2: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [T4: B] :
                ( member(B,T4,C2)
               => topolo1002775350975398744n_open(A,aa(B,set(A),F2,T4)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),C2)))
             => ~ ! [C8: set(B)] :
                    ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C8),C2)
                   => ( aa(set(B),$o,finite_finite2(B),C8)
                     => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),C8))) ) ) ) ) ) ) ).

% compactE_image
tff(fact_5591_compact__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo2193935891317330818ompact(A,bot_bot(set(A))) ) ).

% compact_empty
tff(fact_5592_linorder_OMin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : lattices_Min(A,Less_eq) = lattices_Min(A,Less_eq) ).

% linorder.Min.cong
tff(fact_5593_compact__attains__inf,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,S)
                & ! [Xa2: A] :
                    ( member(A,Xa2,S)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa2) ) ) ) ) ) ).

% compact_attains_inf
tff(fact_5594_compact__attains__sup,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X3: A] :
                ( member(A,X3,S)
                & ! [Xa2: A] :
                    ( member(A,Xa2,S)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa2),X3) ) ) ) ) ) ).

% compact_attains_sup
tff(fact_5595_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ? [X3: A] :
                  ( member(A,X3,S3)
                  & ! [Xa2: A] :
                      ( member(A,Xa2,S3)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Xa2)) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_5596_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ? [X3: A] :
                  ( member(A,X3,S3)
                  & ! [Xa2: A] :
                      ( member(A,Xa2,S3)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa2)),aa(A,B,F2,X3)) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_5597_Cauchy__uniform__iff,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [X4: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X4)
        <=> ! [P5: fun(product_prod(A,A),$o)] :
              ( aa(filter(product_prod(A,A)),$o,aa(fun(product_prod(A,A),$o),fun(filter(product_prod(A,A)),$o),eventually(product_prod(A,A)),P5),topolo7806501430040627800ormity(A))
             => ? [N6: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
                 => ! [M2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M2)
                     => aa(product_prod(A,A),$o,P5,aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,X4,N)),aa(nat,A,X4,M2))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_5598_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))),image2(real,filter(product_prod(complex,complex)),aTP_Lamp_xe(real,filter(product_prod(complex,complex)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_complex_def
tff(fact_5599_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))),image2(real,filter(product_prod(real,real)),aTP_Lamp_xg(real,filter(product_prod(real,real)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_real_def
tff(fact_5600_totally__bounded__def,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S: set(A)] :
          ( topolo6688025880775521714ounded(A,S)
        <=> ! [E6: fun(product_prod(A,A),$o)] :
              ( aa(filter(product_prod(A,A)),$o,aa(fun(product_prod(A,A),$o),fun(filter(product_prod(A,A)),$o),eventually(product_prod(A,A)),E6),topolo7806501430040627800ormity(A))
             => ? [X8: set(A)] :
                  ( aa(set(A),$o,finite_finite2(A),X8)
                  & ! [X: A] :
                      ( member(A,X,S)
                     => ? [Xa3: A] :
                          ( member(A,Xa3,X8)
                          & aa(product_prod(A,A),$o,E6,aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X)) ) ) ) ) ) ) ).

% totally_bounded_def
tff(fact_5601_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P: fun(product_prod(A,A),$o)] :
          ( aa(filter(product_prod(A,A)),$o,aa(fun(product_prod(A,A),$o),fun(filter(product_prod(A,A)),$o),eventually(product_prod(A,A)),P),topolo7806501430040627800ormity(A))
        <=> ? [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
              & ! [X: A,Y4: A] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y4)),E3)
                 => aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),product_Pair(A,A,X),Y4)) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_5602_compact__eq__Heine__Borel,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo2193935891317330818ompact(A,S)
        <=> ! [C4: set(set(A))] :
              ( ( ! [X: set(A)] :
                    ( member(set(A),X,C4)
                   => topolo1002775350975398744n_open(A,X) )
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C4)) )
             => ? [D7: set(set(A))] :
                  ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),D7),C4)
                  & aa(set(set(A)),$o,finite_finite2(set(A)),D7)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),D7)) ) ) ) ) ).

% compact_eq_Heine_Borel
tff(fact_5603_compactI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A)] :
          ( ! [C7: set(set(A))] :
              ( ! [X2: set(A)] :
                  ( member(set(A),X2,C7)
                 => topolo1002775350975398744n_open(A,X2) )
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C7))
               => ? [C9: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C9),C7)
                    & aa(set(set(A)),$o,finite_finite2(set(A)),C9)
                    & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C9)) ) ) )
         => topolo2193935891317330818ompact(A,S3) ) ) ).

% compactI
tff(fact_5604_compactE,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T10: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T10))
           => ( ! [B4: set(A)] :
                  ( member(set(A),B4,T10)
                 => topolo1002775350975398744n_open(A,B4) )
             => ~ ! [T11: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),T11),T10)
                   => ( aa(set(set(A)),$o,finite_finite2(set(A)),T11)
                     => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T11)) ) ) ) ) ) ) ).

% compactE
tff(fact_5605_prod__filter__INF2,axiom,
    ! [C: $tType,B: $tType,A: $tType,J4: set(A),A4: filter(B),B2: fun(A,filter(C))] :
      ( ( J4 != bot_bot(set(A)) )
     => ( prod_filter(B,C,A4,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),B2),J4))) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_xh(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),A4),B2)),J4)) ) ) ).

% prod_filter_INF2
tff(fact_5606_prod__filter__INF1,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A4: fun(A,filter(B)),B2: filter(C)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( prod_filter(B,C,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image2(A,filter(B),A4),I5)),B2) = aa(set(filter(product_prod(B,C))),filter(product_prod(B,C)),complete_Inf_Inf(filter(product_prod(B,C))),aa(set(A),set(filter(product_prod(B,C))),image2(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_xi(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),A4),B2)),I5)) ) ) ).

% prod_filter_INF1
tff(fact_5607_prod__filter__INF,axiom,
    ! [B: $tType,D: $tType,C: $tType,A: $tType,I5: set(A),J4: set(B),A4: fun(A,filter(C)),B2: fun(B,filter(D))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ( J4 != bot_bot(set(B)) )
       => ( prod_filter(C,D,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),A4),I5)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),B2),J4))) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(A),set(filter(product_prod(C,D))),image2(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_xk(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),J4),A4),B2)),I5)) ) ) ) ).

% prod_filter_INF
tff(fact_5608_prod__filter__eq__bot,axiom,
    ! [A: $tType,B: $tType,A4: filter(A),B2: filter(B)] :
      ( ( prod_filter(A,B,A4,B2) = bot_bot(filter(product_prod(A,B))) )
    <=> ( ( A4 = bot_bot(filter(A)) )
        | ( B2 = bot_bot(filter(B)) ) ) ) ).

% prod_filter_eq_bot
tff(fact_5609_prod__filter__mono,axiom,
    ! [A: $tType,B: $tType,F3: filter(A),F8: filter(A),G2: filter(B),G5: filter(B)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F8)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),G2),G5)
       => aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,F3,G2)),prod_filter(A,B,F8,G5)) ) ) ).

% prod_filter_mono
tff(fact_5610_eventually__prod__same,axiom,
    ! [A: $tType,P: fun(product_prod(A,A),$o),F3: filter(A)] :
      ( aa(filter(product_prod(A,A)),$o,aa(fun(product_prod(A,A),$o),fun(filter(product_prod(A,A)),$o),eventually(product_prod(A,A)),P),prod_filter(A,A,F3,F3))
    <=> ? [Q7: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q7),F3)
          & ! [X: A,Y4: A] :
              ( aa(A,$o,Q7,X)
             => ( aa(A,$o,Q7,Y4)
               => aa(product_prod(A,A),$o,P,aa(A,product_prod(A,A),product_Pair(A,A,X),Y4)) ) ) ) ) ).

% eventually_prod_same
tff(fact_5611_eventually__prod__filter,axiom,
    ! [A: $tType,B: $tType,P: fun(product_prod(A,B),$o),F3: filter(A),G2: filter(B)] :
      ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),P),prod_filter(A,B,F3,G2))
    <=> ? [Pf: fun(A,$o),Pg: fun(B,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Pf),F3)
          & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Pg),G2)
          & ! [X: A,Y4: B] :
              ( aa(A,$o,Pf,X)
             => ( aa(B,$o,Pg,Y4)
               => aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),product_Pair(A,B,X),Y4)) ) ) ) ) ).

% eventually_prod_filter
tff(fact_5612_prod__filter__mono__iff,axiom,
    ! [A: $tType,B: $tType,A4: filter(A),B2: filter(B),C2: filter(A),D3: filter(B)] :
      ( ( A4 != bot_bot(filter(A)) )
     => ( ( B2 != bot_bot(filter(B)) )
       => ( aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),prod_filter(A,B,A4,B2)),prod_filter(A,B,C2,D3))
        <=> ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),A4),C2)
            & aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),B2),D3) ) ) ) ) ).

% prod_filter_mono_iff
tff(fact_5613_filterlim__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G2: filter(B),F3: filter(A),G: fun(A,C),H6: filter(C)] :
      ( filterlim(A,B,F2,G2,F3)
     => ( filterlim(A,C,G,H6,F3)
       => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_xl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),F2),G),prod_filter(B,C,G2,H6),F3) ) ) ).

% filterlim_Pair
tff(fact_5614_cauchy__filter__def,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [F3: filter(A)] :
          ( topolo6773858410816713723filter(A,F3)
        <=> aa(filter(product_prod(A,A)),$o,aa(filter(product_prod(A,A)),fun(filter(product_prod(A,A)),$o),ord_less_eq(filter(product_prod(A,A))),prod_filter(A,A,F3,F3)),topolo7806501430040627800ormity(A)) ) ) ).

% cauchy_filter_def
tff(fact_5615_eventually__prod__sequentially,axiom,
    ! [P: fun(product_prod(nat,nat),$o)] :
      ( aa(filter(product_prod(nat,nat)),$o,aa(fun(product_prod(nat,nat),$o),fun(filter(product_prod(nat,nat)),$o),eventually(product_prod(nat,nat)),P),prod_filter(nat,nat,at_top(nat),at_top(nat)))
    <=> ? [N6: nat] :
        ! [M2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M2)
         => ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N)
             => aa(product_prod(nat,nat),$o,P,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,N),M2)) ) ) ) ).

% eventually_prod_sequentially
tff(fact_5616_eventually__prod1,axiom,
    ! [A: $tType,B: $tType,B2: filter(A),P: fun(B,$o),A4: filter(B)] :
      ( ( B2 != bot_bot(filter(A)) )
     => ( aa(filter(product_prod(B,A)),$o,aa(fun(product_prod(B,A),$o),fun(filter(product_prod(B,A)),$o),eventually(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),aTP_Lamp_xm(fun(B,$o),fun(B,fun(A,$o)),P))),prod_filter(B,A,A4,B2))
      <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),A4) ) ) ).

% eventually_prod1
tff(fact_5617_eventually__prod2,axiom,
    ! [A: $tType,B: $tType,A4: filter(A),P: fun(B,$o),B2: filter(B)] :
      ( ( A4 != bot_bot(filter(A)) )
     => ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_xn(fun(B,$o),fun(A,fun(B,$o)),P))),prod_filter(A,B,A4,B2))
      <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),B2) ) ) ).

% eventually_prod2
tff(fact_5618_tendsto__at__iff__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F2: fun(A,B),A3: B,Xa: A,S3: set(A)] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),topolo174197925503356063within(A,Xa,S3))
        <=> ! [X8: fun(nat,A)] :
              ( ! [I4: nat] : member(A,aa(nat,A,X8,I4),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))
             => ( filterlim(nat,A,X8,topolo7230453075368039082e_nhds(A,Xa),at_top(nat))
               => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),comp(A,B,nat,F2),X8),topolo7230453075368039082e_nhds(B,A3),at_top(nat)) ) ) ) ) ).

% tendsto_at_iff_sequentially
tff(fact_5619_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B3: A,A3: A,P: fun(A,$o)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( ! [F5: fun(nat,A)] :
                ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(nat,A,F5,N3))
               => ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F5,N3)),A3)
                 => ( aa(fun(nat,A),$o,order_mono(nat,A),F5)
                   => ( filterlim(nat,A,F5,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(fun(nat,A),fun(nat,$o),aTP_Lamp_wn(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F5)),at_top(nat)) ) ) ) )
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_lessThan(A),A3))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_5620_comp__fun__commute__product__fold,axiom,
    ! [B: $tType,A: $tType,B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),B2)
     => finite6289374366891150609ommute(B,set(product_prod(B,A)),aTP_Lamp_xo(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),B2)) ) ).

% comp_fun_commute_product_fold
tff(fact_5621_surj__fun__eq,axiom,
    ! [B: $tType,C: $tType,A: $tType,F2: fun(B,A),X4: set(B),G1: fun(A,C),G22: fun(A,C)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),X4) = top_top(set(A)) )
     => ( ! [X3: B] :
            ( member(B,X3,X4)
           => ( aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G1),F2),X3) = aa(B,C,aa(fun(B,A),fun(B,C),comp(A,C,B,G22),F2),X3) ) )
       => ( G1 = G22 ) ) ) ).

% surj_fun_eq
tff(fact_5622_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,Na: 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,Na)),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)),Na),F2)) ).

% funpow.simps(2)
tff(fact_5623_funpow__Suc__right,axiom,
    ! [A: $tType,Na: 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,Na)),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)),Na),F2)),F2) ).

% funpow_Suc_right
tff(fact_5624_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F2: fun(A,B),A4: A,B2: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),B2))),aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A4)),aa(A,B,F2,B2))) ) ) ).

% mono_inf
tff(fact_5625_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F2: fun(A,B),A4: A,B2: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A4)),aa(A,B,F2,B2))),aa(A,B,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),A4),B2))) ) ) ).

% mono_sup
tff(fact_5626_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ) ).

% mono_invE
tff(fact_5627_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( aa(fun(nat,A),$o,order_mono(nat,A),X4)
        <=> ! [M2: nat,N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,M2)),aa(nat,A,X4,N)) ) ) ) ).

% incseq_def
tff(fact_5628_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A),I: nat,J: nat] :
          ( aa(fun(nat,A),$o,order_mono(nat,A),F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,I)),aa(nat,A,F2,J)) ) ) ) ).

% incseqD
tff(fact_5629_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
        <=> ! [X: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),aa(A,B,F2,Y4)) ) ) ) ).

% mono_def
tff(fact_5630_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) )
         => aa(fun(A,B),$o,order_mono(A,B),F2) ) ) ).

% monoI
tff(fact_5631_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya)) ) ) ) ).

% monoE
tff(fact_5632_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya)) ) ) ) ).

% monoD
tff(fact_5633_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: fun(nat,A),I: nat] :
          ( aa(fun(nat,A),$o,order_mono(nat,A),A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A4,I)),aa(nat,A,A4,aa(nat,nat,suc,I))) ) ) ).

% incseq_SucD
tff(fact_5634_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X4: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,N2)),aa(nat,A,X4,aa(nat,nat,suc,N2)))
         => aa(fun(nat,A),$o,order_mono(nat,A),X4) ) ) ).

% incseq_SucI
tff(fact_5635_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( aa(fun(nat,A),$o,order_mono(nat,A),F2)
        <=> ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N))) ) ) ).

% incseq_Suc_iff
tff(fact_5636_comp__fun__commute__on__def,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
    <=> ! [X: A,Y4: A] :
          ( member(A,X,S)
         => ( member(A,Y4,S)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),aa(A,fun(B,B),F2,X)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,Y4)) ) ) ) ) ).

% comp_fun_commute_on_def
tff(fact_5637_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Ya: A] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( member(A,Xa,S)
       => ( member(A,Ya,S)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Ya)),aa(A,fun(B,B),F2,Xa)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xa)),aa(A,fun(B,B),F2,Ya)) ) ) ) ) ).

% comp_fun_commute_on.comp_fun_commute_on
tff(fact_5638_comp__fun__commute__on_Ocommute__left__comp,axiom,
    ! [A: $tType,B: $tType,C: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Ya: A,G: fun(C,B)] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( member(A,Xa,S)
       => ( member(A,Ya,S)
         => ( aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Ya)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Xa)),G)) = aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Xa)),aa(fun(C,B),fun(C,B),comp(B,B,C,aa(A,fun(B,B),F2,Ya)),G)) ) ) ) ) ).

% comp_fun_commute_on.commute_left_comp
tff(fact_5639_comp__fun__commute__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,S)
         => ( member(A,Y,S)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y)) ) ) )
     => finite4664212375090638736ute_on(A,B,S,F2) ) ).

% comp_fun_commute_on.intro
tff(fact_5640_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya))
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya) ) ) ) ).

% mono_strict_invE
tff(fact_5641_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Na: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => aa(fun(A,A),$o,order_mono(A,A),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F2)) ) ) ).

% mono_pow
tff(fact_5642_comp__funpow,axiom,
    ! [A: $tType,B: $tType,Na: nat,F2: fun(B,B)] : aa(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B)),aa(nat,fun(fun(fun(A,B),fun(A,B)),fun(fun(A,B),fun(A,B))),compow(fun(fun(A,B),fun(A,B))),Na),comp(B,B,A,F2)) = comp(B,B,A,aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),Na),F2)) ).

% comp_funpow
tff(fact_5643_max__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),M: A,Na: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,M)),aa(A,B,F2,Na)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),M),Na)) ) ) ) ).

% max_of_mono
tff(fact_5644_mono__Suc,axiom,
    aa(fun(nat,nat),$o,order_mono(nat,nat),suc) ).

% mono_Suc
tff(fact_5645_comp__fun__commute_Ointro,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( ! [Y: A,X3: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y))
     => finite6289374366891150609ommute(A,B,F2) ) ).

% comp_fun_commute.intro
tff(fact_5646_comp__fun__commute_Ocomp__fun__commute,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),Ya: A,Xa: A] :
      ( finite6289374366891150609ommute(A,B,F2)
     => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Ya)),aa(A,fun(B,B),F2,Xa)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xa)),aa(A,fun(B,B),F2,Ya)) ) ) ).

% comp_fun_commute.comp_fun_commute
tff(fact_5647_comp__fun__commute__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F2)
    <=> ! [Y4: A,X: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),aa(A,fun(B,B),F2,X)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,Y4)) ) ).

% comp_fun_commute_def
tff(fact_5648_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_5649_mono__add,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A] : aa(fun(A,A),$o,order_mono(A,A),aa(A,fun(A,A),plus_plus(A),A3)) ) ).

% mono_add
tff(fact_5650_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),A4: A,B2: A,Na: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F2),A4)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F2),B2)) ) ) ) ).

% funpow_mono
tff(fact_5651_comp__fun__commute__const,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,B)] : finite6289374366891150609ommute(A,B,aTP_Lamp_xp(fun(B,B),fun(A,fun(B,B)),F2)) ).

% comp_fun_commute_const
tff(fact_5652_mono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Q: fun(A,A)] :
          ( aa(fun(A,A),$o,order_mono(A,A),Q)
         => aa(fun(nat,A),$o,order_mono(nat,A),aTP_Lamp_xq(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_5653_funpow__add,axiom,
    ! [A: $tType,M: nat,Na: 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),Na)),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)),Na),F2)) ).

% funpow_add
tff(fact_5654_filterlim__filtercomap__iff,axiom,
    ! [C: $tType,B: $tType,A: $tType,F2: fun(A,B),G: fun(B,C),G2: filter(C),F3: filter(A)] :
      ( filterlim(A,B,F2,filtercomap(B,C,G,G2),F3)
    <=> filterlim(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G),F2),G2,F3) ) ).

% filterlim_filtercomap_iff
tff(fact_5655_cclfp__lowerbound,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [F2: fun(A,A),A4: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,A4)),A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),order_532582986084564980_cclfp(A,F2)),A4) ) ) ) ).

% cclfp_lowerbound
tff(fact_5656_comp__fun__commute__filter__fold,axiom,
    ! [A: $tType,P: fun(A,$o)] : finite6289374366891150609ommute(A,set(A),aTP_Lamp_lz(fun(A,$o),fun(A,fun(set(A),set(A))),P)) ).

% comp_fun_commute_filter_fold
tff(fact_5657_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),G: fun(A,nat)] :
      ( finite6289374366891150609ommute(A,B,F2)
     => finite6289374366891150609ommute(A,B,aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_mp(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),F2),G)) ) ).

% comp_fun_commute.comp_fun_commute_funpow
tff(fact_5658_mono__times__nat,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => aa(fun(nat,nat),$o,order_mono(nat,nat),aa(nat,fun(nat,nat),times_times(nat),Na)) ) ).

% mono_times_nat
tff(fact_5659_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => aa(fun(A,A),$o,order_mono(A,A),aa(A,fun(A,A),times_times(A),A3)) ) ) ).

% mono_mult
tff(fact_5660_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F2: fun(A,B),M: A,Na: A,M7: B,N5: B] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( ( aa(set(A),set(B),image2(A,B,F2),set_or7035219750837199246ssThan(A,M,Na)) = set_or7035219750837199246ssThan(B,M7,N5) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),M),Na)
             => ( aa(A,B,F2,M) = M7 ) ) ) ) ) ).

% mono_image_least
tff(fact_5661_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A4: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [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)),A4) = aa(B,A,H,aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),A4)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_5662_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F2: fun(A,A),P3: A,K: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),P3),aa(A,A,F2,P3))
           => aa(A,$o,aa(A,fun(A,$o),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_5663_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F2: fun(A,A),P3: A,K: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,P3)),P3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A))),P3) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_5664_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(A,A),I: nat,J: nat,Xa: A,Ya: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(A,A,F2,Xa))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),I),F2),Xa)),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),Ya)) ) ) ) ) ) ).

% funpow_mono2
tff(fact_5665_incseq__bounded,axiom,
    ! [X4: fun(nat,real),B2: real] :
      ( aa(fun(nat,real),$o,order_mono(nat,real),X4)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,X4,I2)),B2)
       => bfun(nat,real,X4,at_top(nat)) ) ) ).

% incseq_bounded
tff(fact_5666_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A4: fun(C,A),I5: set(C)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xr(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A4),I5)))) ) ) ).

% mono_SUP
tff(fact_5667_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ).

% mono_Sup
tff(fact_5668_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ).

% mono_Inf
tff(fact_5669_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F2: fun(A,B),A4: fun(C,A),I5: set(C)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A4),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xr(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4)),I5))) ) ) ).

% mono_INF
tff(fact_5670_Least__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( ? [X2: A] :
                ( member(A,X2,S)
                & ! [Xa4: A] :
                    ( member(A,Xa4,S)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa4) ) )
           => ( ord_Least(B,aa(set(A),fun(B,$o),aTP_Lamp_xs(fun(A,B),fun(set(A),fun(B,$o)),F2),S)) = aa(A,B,F2,ord_Least(A,aTP_Lamp_xt(set(A),fun(A,$o),S))) ) ) ) ) ).

% Least_mono
tff(fact_5671_antimono__funpow,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Q: fun(A,A)] :
          ( aa(fun(A,A),$o,order_mono(A,A),Q)
         => order_antimono(nat,A,aTP_Lamp_xu(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_5672_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X4: fun(nat,A),L4: A,Na: nat] :
          ( aa(fun(nat,A),$o,order_mono(nat,A),X4)
         => ( filterlim(nat,A,X4,topolo7230453075368039082e_nhds(A,L4),at_top(nat))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X4,Na)),L4) ) ) ) ).

% incseq_le
tff(fact_5673_sum_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X3: A,Y: A] :
                ( member(A,X3,A4)
               => ( member(A,Y,A4)
                 => ( ( X3 != Y )
                   => ( ( aa(A,B,H,X3) = aa(A,B,H,Y) )
                     => ( aa(B,C,G,aa(A,B,H,X3)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),aa(set(A),set(B),image2(A,B,H),A4)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H)),A4) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_5674_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M: nat,Na: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),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_5675_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M: nat,Na: nat,F2: fun(A,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),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)),Na),F2),bot_bot(A))) ) ) ) ).

% funpow_decreasing
tff(fact_5676_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),H: fun(A,B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X3: A,Y: A] :
                ( member(A,X3,A4)
               => ( member(A,Y,A4)
                 => ( ( X3 != Y )
                   => ( ( aa(A,B,H,X3) = aa(A,B,H,Y) )
                     => ( aa(B,C,G,aa(A,B,H,X3)) = one_one(C) ) ) ) ) )
           => ( aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),aa(set(A),set(B),image2(A,B,H),A4)) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,B),fun(A,C),comp(B,C,A,G),H)),A4) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_5677_comp__fun__commute__def_H,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite6289374366891150609ommute(A,B,F2)
    <=> finite4664212375090638736ute_on(A,B,top_top(set(A)),F2) ) ).

% comp_fun_commute_def'
tff(fact_5678_incseq__convergent,axiom,
    ! [X4: fun(nat,real),B2: real] :
      ( aa(fun(nat,real),$o,order_mono(nat,real),X4)
     => ( ! [I2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,X4,I2)),B2)
       => ~ ! [L5: real] :
              ( filterlim(nat,real,X4,topolo7230453075368039082e_nhds(real,L5),at_top(nat))
             => ~ ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,X4,I3)),L5) ) ) ) ).

% incseq_convergent
tff(fact_5679_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(A,B,F2,aa(set(A),A,lattic643756798349783984er_Max(A),A4)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image2(A,B,F2),A4)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_5680_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F2: fun(A,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G,aa(A,C,F2,I2))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,aa(fun(C,B),fun(set(C),B),groups7311177749621191930dd_sum(C,B),G),aa(set(A),set(C),image2(A,C,F2),I5))),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),aa(fun(A,C),fun(A,B),comp(C,B,A,G),F2)),I5)) ) ) ) ).

% sum_image_le
tff(fact_5681_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: 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,Na))) = 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),Na))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_5682_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Na: 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,Na))) = 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),Na))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_5683_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Na))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_5684_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Na))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_5685_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) = 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,minus_minus(nat,Na),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_5686_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_5687_mono__cSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A4: fun(C,A),I5: set(C)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( condit941137186595557371_above(A,aa(set(C),set(A),image2(C,A,A4),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xv(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A4),I5)))) ) ) ) ) ).

% mono_cSUP
tff(fact_5688_mono__cSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( condit941137186595557371_above(A,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ) ) ).

% mono_cSup
tff(fact_5689_mono__cInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( condit1013018076250108175_below(A,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ) ).

% mono_cInf
tff(fact_5690_mono__cINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(B)
        & condit1219197933456340205attice(A) )
     => ! [F2: fun(A,B),A4: fun(C,A),I5: set(C)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( condit1013018076250108175_below(A,aa(set(C),set(A),image2(C,A,A4),I5))
           => ( ( I5 != bot_bot(set(C)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A4),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xv(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4)),I5))) ) ) ) ) ).

% mono_cINF
tff(fact_5691_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: 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_xw(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_5692_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Na: 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_xw(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_5693_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_xw(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_5694_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Na: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_xw(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,Na)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_5695_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(fun(nat,nat),$o,order_mono(nat,nat),aTP_Lamp_xx(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_5696_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = 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,minus_minus(nat,Na),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_5697_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Na: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,Na)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,Na),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_5698_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat))))
         => ( aa(fun(nat,A),$o,order_mono(nat,A),F2)
           => ( ! [N2: nat] :
                  ( ( aa(nat,A,F2,N2) = aa(nat,A,F2,aa(nat,nat,suc,N2)) )
                 => ( aa(nat,A,F2,aa(nat,nat,suc,N2)) = aa(nat,A,F2,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) )
             => ? [N8: nat] :
                  ( ! [N3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),N8)
                     => ! [M3: nat] :
                          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M3),N8)
                         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M3),N3)
                           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,M3)),aa(nat,A,F2,N3)) ) ) )
                  & ! [N3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N8),N3)
                     => ( aa(nat,A,F2,N8) = aa(nat,A,F2,N3) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_5699_tendsto__at__left__sequentially,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [B3: A,A3: A,X4: fun(A,B),L4: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( ! [S2: fun(nat,A)] :
                ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,S2,N3)),A3)
               => ( ! [N3: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),aa(nat,A,S2,N3))
                 => ( aa(fun(nat,A),$o,order_mono(nat,A),S2)
                   => ( filterlim(nat,A,S2,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
                     => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wq(fun(A,B),fun(fun(nat,A),fun(nat,B)),X4),S2),topolo7230453075368039082e_nhds(B,L4),at_top(nat)) ) ) ) )
           => filterlim(A,B,X4,topolo7230453075368039082e_nhds(B,L4),topolo174197925503356063within(A,A3,aa(A,set(A),set_ord_lessThan(A),A3))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_5700_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_5701_continuous__at__Sup__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Sup_Sup(A),S),aa(A,set(A),set_ord_lessThan(A),aa(set(A),A,complete_Sup_Sup(A),S))),F2)
           => ( ( S != bot_bot(set(A)) )
             => ( condit941137186595557371_above(A,S)
               => ( aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),S)) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),S)) ) ) ) ) ) ) ).

% continuous_at_Sup_mono
tff(fact_5702_continuous__at__Inf__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & topolo1944317154257567458pology(A)
        & condit6923001295902523014norder(B)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(set(A),A,complete_Inf_Inf(A),S),aa(A,set(A),set_ord_greaterThan(A),aa(set(A),A,complete_Inf_Inf(A),S))),F2)
           => ( ( S != bot_bot(set(A)) )
             => ( condit1013018076250108175_below(A,S)
               => ( aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),S)) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),S)) ) ) ) ) ) ) ).

% continuous_at_Inf_mono
tff(fact_5703_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y3: set(B),X4: set(A),F3: filter(B),F2: fun(A,C)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),Y3)),X4)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aTP_Lamp_xy(set(B),fun(B,$o),Y3)),F3)
       => ( aa(filter(A),filter(C),aa(fun(A,C),fun(filter(A),filter(C)),map_filter_on(A,C,X4),F2),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),map_filter_on(B,A,Y3),G),F3)) = aa(filter(B),filter(C),aa(fun(B,C),fun(filter(B),filter(C)),map_filter_on(B,C,Y3),aa(fun(B,A),fun(B,C),comp(A,C,B,F2),G)),F3) ) ) ) ).

% map_filter_on_comp
tff(fact_5704_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys2 )
    <=> ? [F7: fun(nat,nat)] :
          ( aa(fun(nat,nat),$o,order_mono(nat,nat),F7)
          & ( aa(set(nat),set(nat),image2(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)),Ys2)) )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
             => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys2),aa(nat,nat,F7,I4)) ) )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))),aa(list(A),nat,size_size(list(A)),Xs))
             => ( ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) )
              <=> ( aa(nat,nat,F7,I4) = aa(nat,nat,F7,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_5705_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),K)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),K),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))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ya(set(product_prod(A,A)),fun(nat,$o),R))))) ) ) ).

% relpow_finite_bounded1
tff(fact_5706_finite__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Na: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Na),R)) ) ) ).

% finite_relpow
tff(fact_5707_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F2: fun(A,C),G: fun(D,B)] : aa(fun(A,C),fun(A,set(B)),comp(C,set(B),A,aTP_Lamp_yb(C,set(B))),F2) = aa(fun(A,set(D)),fun(A,set(B)),comp(set(D),set(B),A,image2(D,B,G)),aTP_Lamp_yc(A,set(D))) ).

% empty_natural
tff(fact_5708_mono__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A4: set(A),B2: set(A)] :
      ( aa(fun(set(A),set(B)),$o,order_mono(set(A),set(B)),F2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),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,A4)),aa(set(A),set(B),F2,B2))),aa(set(A),set(B),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))) ) ).

% mono_Un
tff(fact_5709_mono__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(set(A),set(B)),A4: set(A),B2: set(A)] :
      ( aa(fun(set(A),set(B)),$o,order_mono(set(A),set(B)),F2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),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)),A4),B2))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F2,A4)),aa(set(A),set(B),F2,B2))) ) ).

% mono_Int
tff(fact_5710_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(A,fun(B,B)),G: fun(C,A)] :
      ( finite6289374366891150609ommute(A,B,F2)
     => finite6289374366891150609ommute(C,B,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F2),G)) ) ).

% comp_fun_commute.comp_comp_fun_commute
tff(fact_5711_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% remdups_adj_length
tff(fact_5712_relpow__0__I,axiom,
    ! [A: $tType,Xa: A,R: set(product_prod(A,A))] : member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Xa),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R)) ).

% relpow_0_I
tff(fact_5713_relpow__0__E,axiom,
    ! [A: $tType,Xa: A,Ya: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),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))
     => ( Xa = Ya ) ) ).

% relpow_0_E
tff(fact_5714_relpow__E,axiom,
    ! [A: $tType,Xa: A,Z: A,Na: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),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))),Na),R))
     => ( ( ( Na = zero_zero(nat) )
         => ( Xa != Z ) )
       => ~ ! [Y: A,M4: nat] :
              ( ( Na = aa(nat,nat,suc,M4) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),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))
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),Z),R) ) ) ) ) ).

% relpow_E
tff(fact_5715_relpow__E2,axiom,
    ! [A: $tType,Xa: A,Z: A,Na: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),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))),Na),R))
     => ( ( ( Na = zero_zero(nat) )
         => ( Xa != Z ) )
       => ~ ! [Y: A,M4: nat] :
              ( ( Na = aa(nat,nat,suc,M4) )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Y),R)
               => ~ 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_5716_eventually__map__filter__on,axiom,
    ! [B: $tType,A: $tType,X4: set(A),F3: filter(A),P: fun(B,$o),F2: fun(A,B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),X4)),F3)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),map_filter_on(A,B,X4),F2),F3))
      <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_yd(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),X4),P),F2)),F3) ) ) ).

% eventually_map_filter_on
tff(fact_5717_relpow__empty,axiom,
    ! [A: $tType,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( 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))),Na),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_5718_prod_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B2: set(set(A)),G: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,B2)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [A13: set(A)] :
                ( member(set(A),A13,B2)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B2)
                   => ( ( A13 != A24 )
                     => ! [X3: A] :
                          ( member(A,X3,A13)
                         => ( member(A,X3,A24)
                           => ( aa(A,B,G,X3) = one_one(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B2)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),B2) ) ) ) ) ).

% prod.Union_comp
tff(fact_5719_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),G),A4) = finite_fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,plus_plus(A)),G),zero_zero(A),A4) ) ).

% sum.eq_fold
tff(fact_5720_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F2: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,G),top_top(set(C)))),S)
       => finite4664212375090638736ute_on(C,B,R,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F2),G)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_5721_remdups__adj__adjacent,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs)))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I)) ) ) ).

% remdups_adj_adjacent
tff(fact_5722_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S: set(int)] :
      ( ~ aa(set(int),$o,finite_finite2(int),S)
    <=> ~ aa(set(nat),$o,finite_finite2(nat),aa(set(int),set(nat),image2(int,nat,aa(fun(int,int),fun(int,nat),comp(int,nat,int,nat2),abs_abs(int))),S)) ) ).

% infinite_int_iff_infinite_nat_abs
tff(fact_5723_relpow__fun__conv,axiom,
    ! [A: $tType,A3: A,B3: A,Na: nat,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),Na),R))
    <=> ? [F7: fun(nat,A)] :
          ( ( aa(nat,A,F7,zero_zero(nat)) = A3 )
          & ( aa(nat,A,F7,Na) = B3 )
          & ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),Na)
             => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F7,I4)),aa(nat,A,F7,aa(nat,nat,suc,I4))),R) ) ) ) ).

% relpow_fun_conv
tff(fact_5724_prod_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C2: set(set(A)),G: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,C2)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [X3: set(A)] :
                ( member(set(A),X3,C2)
               => ! [Xa4: set(A)] :
                    ( member(set(A),Xa4,C2)
                   => ( ( X3 != Xa4 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7121269368397514597t_prod(set(A),B)),groups7121269368397514597t_prod(A,B)),G),C2) ) ) ) ) ).

% prod.Union_disjoint
tff(fact_5725_sup__SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),B2: B,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),B2),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F2),B2,A4) ) ) ) ).

% sup_SUP_fold_sup
tff(fact_5726_inf__INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),B2: B,F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(B,B,aa(B,fun(B,B),inf_inf(B),B2),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F2),B2,A4) ) ) ) ).

% inf_INF_fold_inf
tff(fact_5727_sum_OUnion__comp,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B2: set(set(A)),G: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,B2)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [A13: set(A)] :
                ( member(set(A),A13,B2)
               => ! [A24: set(A)] :
                    ( member(set(A),A24,B2)
                   => ( ( A13 != A24 )
                     => ! [X3: A] :
                          ( member(A,X3,A13)
                         => ( member(A,X3,A24)
                           => ( aa(A,B,G,X3) = zero_zero(B) ) ) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B2)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G),B2) ) ) ) ) ).

% sum.Union_comp
tff(fact_5728_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))) ) ).

% remdups_adj_length_ge1
tff(fact_5729_relpow__finite__bounded,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),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))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ye(set(product_prod(A,A)),fun(nat,$o),R))))) ) ).

% relpow_finite_bounded
tff(fact_5730_SUP__fold__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,sup_sup(B)),F2),bot_bot(B),A4) ) ) ) ).

% SUP_fold_sup
tff(fact_5731_INF__fold__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4)) = finite_fold(A,B,aa(fun(A,B),fun(A,fun(B,B)),comp(B,fun(B,B),A,inf_inf(B)),F2),top_top(B),A4) ) ) ) ).

% INF_fold_inf
tff(fact_5732_sum_OUnion__disjoint,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C2: set(set(A)),G: fun(A,B)] :
          ( ! [X3: set(A)] :
              ( member(set(A),X3,C2)
             => aa(set(A),$o,finite_finite2(A),X3) )
         => ( ! [X3: set(A)] :
                ( member(set(A),X3,C2)
               => ! [Xa4: set(A)] :
                    ( member(set(A),Xa4,C2)
                   => ( ( X3 != Xa4 )
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X3),Xa4) = bot_bot(set(A)) ) ) ) )
           => ( aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),G),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2)) = aa(set(set(A)),B,aa(fun(A,B),fun(set(set(A)),B),aa(fun(fun(A,B),fun(set(A),B)),fun(fun(A,B),fun(set(set(A)),B)),comp(fun(set(A),B),fun(set(set(A)),B),fun(A,B),groups7311177749621191930dd_sum(set(A),B)),groups7311177749621191930dd_sum(A,B)),G),C2) ) ) ) ) ).

% sum.Union_disjoint
tff(fact_5733_comp__fun__commute__Pow__fold,axiom,
    ! [A: $tType] : finite6289374366891150609ommute(A,set(set(A)),aTP_Lamp_lx(A,fun(set(set(A)),set(set(A))))) ).

% comp_fun_commute_Pow_fold
tff(fact_5734_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F2: fun(A,real),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ud(fun(A,real),fun(A,$o),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_5735_ntrancl__def,axiom,
    ! [A: $tType,Na: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,Na,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_yf(nat,fun(nat,$o),Na)))) ).

% ntrancl_def
tff(fact_5736_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(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))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ya(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_5737_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F2: fun(nat,real),G: fun(nat,nat)] :
      ( ! [X3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X3))
     => ( aa(fun(nat,real),$o,order_mono(nat,real),F2)
       => ( order_strict_mono(nat,nat,G)
         => ( bfun(nat,real,aa(fun(nat,nat),fun(nat,real),aTP_Lamp_yg(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_5738_ntrancl__Zero,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : transitive_ntrancl(A,zero_zero(nat),R) = R ).

% ntrancl_Zero
tff(fact_5739_strict__mono__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
         => aa(fun(A,B),$o,order_mono(A,B),F2) ) ) ).

% strict_mono_mono
tff(fact_5740_infinite__enumerate,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => ? [R3: fun(nat,nat)] :
          ( order_strict_mono(nat,nat,R3)
          & ! [N3: nat] : member(nat,aa(nat,nat,R3,N3),S) ) ) ).

% infinite_enumerate
tff(fact_5741_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya)) ) ) ) ).

% strict_monoD
tff(fact_5742_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,F2,Y)) )
         => order_strict_mono(A,B,F2) ) ) ).

% strict_monoI
tff(fact_5743_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F2: fun(A,B)] :
          ( order_strict_mono(A,B,F2)
        <=> ! [X: A,Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y4)
             => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),aa(A,B,F2,Y4)) ) ) ) ).

% strict_mono_def
tff(fact_5744_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya) ) ) ) ).

% strict_mono_less
tff(fact_5745_strict__mono__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( order_strict_mono(A,B,F2)
         => ( ( aa(A,B,F2,Xa) = aa(A,B,F2,Ya) )
          <=> ( Xa = Ya ) ) ) ) ).

% strict_mono_eq
tff(fact_5746_strict__mono__imp__increasing,axiom,
    ! [F2: fun(nat,nat),Na: nat] :
      ( order_strict_mono(nat,nat,F2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(nat,nat,F2,Na)) ) ).

% strict_mono_imp_increasing
tff(fact_5747_strict__mono__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [R2: fun(A,B),M: A,Na: A] :
          ( order_strict_mono(A,B,R2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),Na)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,R2,M)),aa(A,B,R2,Na)) ) ) ) ).

% strict_mono_leD
tff(fact_5748_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( order_strict_mono(A,B,F2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya) ) ) ) ).

% strict_mono_less_eq
tff(fact_5749_trancl__mono,axiom,
    ! [A: $tType,P3: product_prod(A,A),R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P3,transitive_trancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S3)
       => member(product_prod(A,A),P3,transitive_trancl(A,S3)) ) ) ).

% trancl_mono
tff(fact_5750_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F2: fun(nat,A)] :
          ( order_strict_mono(nat,A,F2)
        <=> ! [N: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,F2,N)),aa(nat,A,F2,aa(nat,nat,suc,N))) ) ) ).

% strict_mono_Suc_iff
tff(fact_5751_trancl__power,axiom,
    ! [A: $tType,P3: product_prod(A,A),R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),P3,transitive_trancl(A,R))
    <=> ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
          & 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))),N),R)) ) ) ).

% trancl_power
tff(fact_5752_strict__mono__enumerate,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => order_strict_mono(nat,nat,infini527867602293511546merate(nat,S)) ) ).

% strict_mono_enumerate
tff(fact_5753_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] :
                ( ~ member(nat,N2,aa(set(nat),set(nat),image2(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_yh(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2))
            <=> summable(A,F2) ) ) ) ) ).

% summable_mono_reindex
tff(fact_5754_sums__mono__reindex,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(nat,nat),F2: fun(nat,A),C3: A] :
          ( order_strict_mono(nat,nat,G)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,aa(set(nat),set(nat),image2(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_yh(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2),C3)
            <=> sums(A,F2,C3) ) ) ) ) ).

% sums_mono_reindex
tff(fact_5755_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] :
                ( ~ member(nat,N2,aa(set(nat),set(nat),image2(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_yi(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),G),F2)) = suminf(A,F2) ) ) ) ) ).

% suminf_mono_reindex
tff(fact_5756_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F2: fun(nat,A),G: fun(nat,nat)] :
          ( ! [X3: nat,Y: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Y)
             => aa(real,$o,aa(real,fun(real,$o),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_yj(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_5757_coinduct3__mono__lemma,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [F2: fun(A,set(B)),X4: set(B),B2: set(B)] :
          ( aa(fun(A,set(B)),$o,order_mono(A,set(B)),F2)
         => aa(fun(A,set(B)),$o,order_mono(A,set(B)),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_yk(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),F2),X4),B2)) ) ) ).

% coinduct3_mono_lemma
tff(fact_5758_compact__imp__fip__image,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),I5: set(B),F2: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S3)
         => ( ! [I2: B] :
                ( member(B,I2,I5)
               => topolo7761053866217962861closed(A,aa(B,set(A),F2,I2)) )
           => ( ! [I6: set(B)] :
                  ( aa(set(B),$o,finite_finite2(B),I6)
                 => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),I6),I5)
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),I6))) != bot_bot(set(A)) ) ) )
             => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image2(B,set(A),F2),I5))) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip_image
tff(fact_5759_inj__sgn__power,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => inj_on(real,real,aTP_Lamp_mr(nat,fun(real,real),Na),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_5760_inj__on__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B)] : inj_on(A,B,F2,bot_bot(set(A))) ).

% inj_on_empty
tff(fact_5761_closed__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo7761053866217962861closed(A,bot_bot(set(A))) ) ).

% closed_empty
tff(fact_5762_closed__Un,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T3: set(A)] :
          ( topolo7761053866217962861closed(A,S)
         => ( topolo7761053866217962861closed(A,T3)
           => topolo7761053866217962861closed(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) ) ) ) ).

% closed_Un
tff(fact_5763_closed__singleton,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [A3: A] : topolo7761053866217962861closed(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ).

% closed_singleton
tff(fact_5764_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_5765_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_yl(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_5766_closed__UN,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [A4: set(A),B2: fun(A,set(B))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => topolo7761053866217962861closed(B,aa(A,set(B),B2,X3)) )
           => topolo7761053866217962861closed(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),B2),A4))) ) ) ) ).

% closed_UN
tff(fact_5767_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: A,A4: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))
    <=> ( inj_on(A,B,F2,A4)
        & ~ member(B,aa(A,B,F2,A3),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_5768_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(B,A),T3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(B),set(A),image2(B,A,F2),T3))
    <=> ? [U4: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),U4),T3)
          & inj_on(B,A,F2,U4)
          & ( S = aa(set(B),set(A),image2(B,A,F2),U4) ) ) ) ).

% subset_image_inj
tff(fact_5769_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B2: set(A),A3: A,A4: set(A)] :
      ( inj_on(A,B,F2,B2)
     => ( member(A,A3,B2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => ( member(B,aa(A,B,F2,A3),aa(set(A),set(B),image2(A,B,F2),A4))
          <=> member(A,A3,A4) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_5770_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C2: set(A),A4: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,C2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
         => ( ( aa(set(A),set(B),image2(A,B,F2),A4) = aa(set(A),set(B),image2(A,B,F2),B2) )
          <=> ( A4 = B2 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_5771_finite__image__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image2(A,B,F2),A4))
      <=> aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% finite_image_iff
tff(fact_5772_finite__imageD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A4))
     => ( inj_on(B,A,F2,A4)
       => aa(set(B),$o,finite_finite2(B),A4) ) ) ).

% finite_imageD
tff(fact_5773_inj__img__insertE,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),Xa: B,B2: set(B)] :
      ( inj_on(A,B,F2,A4)
     => ( ~ member(B,Xa,B2)
       => ( ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),B2) = aa(set(A),set(B),image2(A,B,F2),A4) )
         => ~ ! [X7: A,A6: set(A)] :
                ( ~ member(A,X7,A6)
               => ( ( A4 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X7),A6) )
                 => ( ( Xa = aa(A,B,F2,X7) )
                   => ( B2 != aa(set(A),set(B),image2(A,B,F2),A6) ) ) ) ) ) ) ) ).

% inj_img_insertE
tff(fact_5774_inj__on__Un__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
     => ( ( aa(set(A),set(B),image2(A,B,F2),A4) = aa(set(A),set(B),image2(A,B,F2),B2) )
      <=> ( A4 = B2 ) ) ) ).

% inj_on_Un_image_eq_iff
tff(fact_5775_card__image,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A4)) = aa(set(A),nat,finite_card(A),A4) ) ) ).

% card_image
tff(fact_5776_inj__on__image__Fpow,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F2,A4)
     => inj_on(set(A),set(B),image2(A,B,F2),finite_Fpow(A,A4)) ) ).

% inj_on_image_Fpow
tff(fact_5777_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B2: set(A),A4: set(A)] :
      ( inj_on(A,B,F2,B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),aa(set(A),set(B),image2(A,B,F2),B2)) ) ) ).

% inj_on_strict_subset
tff(fact_5778_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
       => inj_on(A,B,F2,B2) ) ) ).

% inj_on_subset
tff(fact_5779_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B2: set(A),A4: set(A)] :
      ( inj_on(A,B,F2,B2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
       => inj_on(A,B,F2,A4) ) ) ).

% subset_inj_on
tff(fact_5780_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A4: set(A),F2: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
             => ( member(A,X3,A4)
               => ( member(A,Y,A4)
                 => ( aa(A,B,F2,X3) != aa(A,B,F2,Y) ) ) ) )
         => ( ! [X3: A,Y: A] :
                ( member(A,X3,A4)
               => ( member(A,Y,A4)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X3) ) ) )
           => inj_on(A,B,F2,A4) ) ) ) ).

% linorder_inj_onI
tff(fact_5781_continuous__on__closed__Un,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),Ta: set(A),F2: fun(A,B)] :
          ( topolo7761053866217962861closed(A,S3)
         => ( topolo7761053866217962861closed(A,Ta)
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ( topolo81223032696312382ous_on(A,B,Ta,F2)
               => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),Ta),F2) ) ) ) ) ) ).

% continuous_on_closed_Un
tff(fact_5782_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(B,A),D3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( inj_on(B,A,F2,D3)
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_ym(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A4),F2),D3))) ) ) ).

% finite_inverse_image_gen
tff(fact_5783_finite__imp__closed,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => topolo7761053866217962861closed(A,S) ) ) ).

% finite_imp_closed
tff(fact_5784_closed__insert,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [S: set(A),A3: A] :
          ( topolo7761053866217962861closed(A,S)
         => topolo7761053866217962861closed(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),S)) ) ) ).

% closed_insert
tff(fact_5785_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F2: fun(A,B)] :
          ( ! [X3: A,Y: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
             => ( aa(A,B,F2,X3) != aa(A,B,F2,Y) ) )
         => inj_on(A,B,F2,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_5786_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,A4: set(A)] :
          ( ( A3 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A4) ) ) ).

% inj_on_mult
tff(fact_5787_inj__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Na: 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)),Na),F2),top_top(set(A))) ) ).

% inj_fn
tff(fact_5788_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( inj_on(B,A,F2,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_yn(set(A),fun(fun(B,A),fun(B,$o)),A4),F2))) ) ) ).

% finite_inverse_image
tff(fact_5789_continuous__on__If,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),Ta: set(A),F2: fun(A,B),G: fun(A,B),P: fun(A,$o)] :
          ( topolo7761053866217962861closed(A,S3)
         => ( topolo7761053866217962861closed(A,Ta)
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ( topolo81223032696312382ous_on(A,B,Ta,G)
               => ( ! [X3: A] :
                      ( member(A,X3,S3)
                     => ( ~ aa(A,$o,P,X3)
                       => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
                 => ( ! [X3: A] :
                        ( member(A,X3,Ta)
                       => ( aa(A,$o,P,X3)
                         => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) ) )
                   => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),Ta),aa(fun(A,$o),fun(A,B),aa(fun(A,B),fun(fun(A,$o),fun(A,B)),aTP_Lamp_yo(fun(A,B),fun(fun(A,B),fun(fun(A,$o),fun(A,B))),F2),G),P)) ) ) ) ) ) ) ) ).

% continuous_on_If
tff(fact_5790_continuous__on__cases,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S3: set(A),Ta: set(A),F2: fun(A,B),G: fun(A,B),P: fun(A,$o)] :
          ( topolo7761053866217962861closed(A,S3)
         => ( topolo7761053866217962861closed(A,Ta)
           => ( topolo81223032696312382ous_on(A,B,S3,F2)
             => ( topolo81223032696312382ous_on(A,B,Ta,G)
               => ( ! [X3: A] :
                      ( ( ( member(A,X3,S3)
                          & ~ aa(A,$o,P,X3) )
                        | ( member(A,X3,Ta)
                          & aa(A,$o,P,X3) ) )
                     => ( aa(A,B,F2,X3) = aa(A,B,G,X3) ) )
                 => topolo81223032696312382ous_on(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),Ta),aa(fun(A,$o),fun(A,B),aa(fun(A,B),fun(fun(A,$o),fun(A,B)),aTP_Lamp_yo(fun(A,B),fun(fun(A,B),fun(fun(A,$o),fun(A,B))),F2),G),P)) ) ) ) ) ) ) ).

% continuous_on_cases
tff(fact_5791_finite__UNIV__inj__surj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( inj_on(A,A,F2,top_top(set(A)))
       => ( aa(set(A),set(A),image2(A,A,F2),top_top(set(A))) = top_top(set(A)) ) ) ) ).

% finite_UNIV_inj_surj
tff(fact_5792_finite__UNIV__surj__inj,axiom,
    ! [A: $tType,F2: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( ( aa(set(A),set(A),image2(A,A,F2),top_top(set(A))) = top_top(set(A)) )
       => inj_on(A,A,F2,top_top(set(A))) ) ) ).

% finite_UNIV_surj_inj
tff(fact_5793_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),aa(set(A),set(B),image2(A,B,F2),B2))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2) ) ) ).

% inj_image_subset_iff
tff(fact_5794_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A4: set(A),A14: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ? [F7: fun(A,B)] :
            ( inj_on(A,B,F7,A4)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A4)),A14) )
      <=> ? [G6: fun(B,A)] : aa(set(B),set(A),image2(B,A,G6),A14) = A4 ) ) ).

% inj_on_iff_surj
tff(fact_5795_finite__surj__inj,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),image2(A,A,F2),A4))
       => inj_on(A,A,F2,A4) ) ) ).

% finite_surj_inj
tff(fact_5796_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B2: set(B)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),B2)
       => ( aa(set(B),$o,finite_finite2(B),B2)
         => aa(set(A),$o,finite_finite2(A),A4) ) ) ) ).

% inj_on_finite
tff(fact_5797_endo__inj__surj,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image2(A,A,F2),A4)),A4)
       => ( inj_on(A,A,F2,A4)
         => ( aa(set(A),set(A),image2(A,A,F2),A4) = A4 ) ) ) ) ).

% endo_inj_surj
tff(fact_5798_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C2: set(A),A4: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,C2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
         => ( aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),aa(set(A),set(B),image2(A,B,F2),B2)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_5799_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),C2: set(A),A4: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,C2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A4),B2)),C2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),C2)
         => ( aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),minus_minus(set(A),A4),B2)) = aa(set(B),set(B),minus_minus(set(B),aa(set(A),set(B),image2(A,B,F2),A4)),aa(set(A),set(B),image2(A,B,F2),B2)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_5800_inj__on__iff__eq__card,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( inj_on(A,B,F2,A4)
      <=> ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A4)) = aa(set(A),nat,finite_card(A),A4) ) ) ) ).

% inj_on_iff_eq_card
tff(fact_5801_eq__card__imp__inj__on,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ( aa(set(B),nat,finite_card(B),aa(set(A),set(B),image2(A,B,F2),A4)) = aa(set(A),nat,finite_card(A),A4) )
       => inj_on(A,B,F2,A4) ) ) ).

% eq_card_imp_inj_on
tff(fact_5802_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image2(B,A,F2),A4))),aa(set(B),nat,finite_card(B),A4))
     => ~ inj_on(B,A,F2,A4) ) ).

% pigeonhole
tff(fact_5803_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A3: A,Xa: A,B3: A,F2: fun(A,B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),B3)
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A3,B3),F2)
             => ( inj_on(A,B,F2,set_or1337092689740270186AtMost(A,A3,B3))
               => ( ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,A3)),aa(A,B,F2,Xa))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,B3)) )
                  | ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,B3)),aa(A,B,F2,Xa))
                    & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Xa)),aa(A,B,F2,A3)) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_5804_fold__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A4: set(A),F2: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G,A4)
     => ( finite_fold(B,C,F2,Z,aa(set(A),set(B),image2(A,B,G),A4)) = finite_fold(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F2),G),Z,A4) ) ) ).

% fold_image
tff(fact_5805_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),Xa: B,B2: set(A)] :
      ( inj_on(A,B,F2,A4)
     => ( member(B,Xa,aa(set(A),set(B),image2(A,B,F2),A4))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => member(A,the_inv_into(A,B,A4,F2,Xa),B2) ) ) ) ).

% the_inv_into_into
tff(fact_5806_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X4: fun($o,A),Y3: fun($o,A)] :
          ( aa(fun($o,A),$o,aa(fun($o,A),fun(fun($o,A),$o),ord_less_eq(fun($o,A)),X4),Y3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,X4,$false)),aa($o,A,Y3,$false))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,X4,$true)),aa($o,A,Y3,$true)) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_5807_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B)),F2: fun(B,C)] :
      ( ! [I2: A,J2: A] :
          ( member(A,I2,I5)
         => ( member(A,J2,I5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A4,I2)),aa(A,set(B),A4,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A4,J2)),aa(A,set(B),A4,I2)) ) ) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => inj_on(B,C,F2,aa(A,set(B),A4,I2)) )
       => inj_on(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_5808_inj__on__INTER,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),F2: fun(B,C),A4: fun(A,set(B))] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => inj_on(B,C,F2,aa(A,set(B),A4,I2)) )
       => inj_on(B,C,F2,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5))) ) ) ).

% inj_on_INTER
tff(fact_5809_closed__Collect__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F2)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo7761053866217962861closed(A,aa(fun(A,$o),set(A),collect(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_yp(fun(A,B),fun(fun(A,B),fun(A,$o)),F2),G))) ) ) ) ).

% closed_Collect_le
tff(fact_5810_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B2: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),B2)
       => ( inj_on(B,A,G,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),B2)),A4)
           => ( aa(set(A),$o,finite_finite2(A),A4)
             => ( aa(set(B),$o,finite_finite2(B),B2)
               => ( aa(set(A),nat,finite_card(A),A4) = aa(set(B),nat,finite_card(B),B2) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_5811_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S: set(A),T3: set(B),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => ( aa(set(B),$o,finite_finite2(B),T3)
       => ( ( aa(set(A),nat,finite_card(A),S) = aa(set(B),nat,finite_card(B),T3) )
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),S)),T3)
           => ( ! [X: B] :
                  ( member(B,X,T3)
                 => ? [Xa3: A] :
                      ( member(A,Xa3,S)
                      & ( aa(A,B,F2,Xa3) = X ) ) )
            <=> inj_on(A,B,F2,S) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_5812_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),uminus_uminus(set(A)),A4))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image2(A,B,F2),A4))) ) ).

% inj_image_Compl_subset
tff(fact_5813_t3__space,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [S: set(A),Ya: A] :
          ( topolo7761053866217962861closed(A,S)
         => ( ~ member(A,Ya,S)
           => ? [U5: set(A),V5: set(A)] :
                ( topolo1002775350975398744n_open(A,U5)
                & topolo1002775350975398744n_open(A,V5)
                & member(A,Ya,U5)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),V5)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V5) = bot_bot(set(A)) ) ) ) ) ) ).

% t3_space
tff(fact_5814_t4__space,axiom,
    ! [A: $tType] :
      ( topological_t4_space(A)
     => ! [S: set(A),T3: set(A)] :
          ( topolo7761053866217962861closed(A,S)
         => ( topolo7761053866217962861closed(A,T3)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T3) = bot_bot(set(A)) )
             => ? [U5: set(A),V5: set(A)] :
                  ( topolo1002775350975398744n_open(A,U5)
                  & topolo1002775350975398744n_open(A,V5)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),U5)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),V5)
                  & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V5) = bot_bot(set(A)) ) ) ) ) ) ) ).

% t4_space
tff(fact_5815_inj__on__disjoint__Un,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),G: fun(A,B),B2: set(A)] :
      ( inj_on(A,B,F2,A4)
     => ( inj_on(A,B,G,B2)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),aa(set(A),set(B),image2(A,B,G),B2)) = bot_bot(set(B)) )
         => inj_on(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_yq(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A4),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ) ) ) ).

% inj_on_disjoint_Un
tff(fact_5816_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(A,B),C2: set(A),A4: set(C),B2: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F2,C2)
     => ( ! [X3: C] :
            ( member(C,X3,A4)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(C,set(A),B2,X3)),C2) )
       => ( member(C,J,A4)
         => ( aa(set(A),set(B),image2(A,B,F2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),B2),A4))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_yr(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F2),B2)),A4)) ) ) ) ) ).

% image_INT
tff(fact_5817_nhds__closed,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Xa: A,A4: set(A)] :
          ( member(A,Xa,A4)
         => ( topolo1002775350975398744n_open(A,A4)
           => ? [A6: set(A)] :
                ( member(A,Xa,A6)
                & topolo7761053866217962861closed(A,A6)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A6),A4)
                & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ys(set(A),fun(A,$o),A6)),topolo7230453075368039082e_nhds(A,Xa)) ) ) ) ) ).

% nhds_closed
tff(fact_5818_continuous__on__closed__Union,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [I5: set(A),U2: fun(A,set(B)),F2: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => topolo7761053866217962861closed(B,aa(A,set(B),U2,I2)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => topolo81223032696312382ous_on(B,C,aa(A,set(B),U2,I2),F2) )
             => topolo81223032696312382ous_on(B,C,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),U2),I5)),F2) ) ) ) ) ).

% continuous_on_closed_Union
tff(fact_5819_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B2))
         => ? [F5: fun(A,B)] :
              ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F5),A4)),B2)
              & inj_on(A,B,F5,A4) ) ) ) ) ).

% card_le_inj
tff(fact_5820_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B2: set(B)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),B2)
       => ( aa(set(B),$o,finite_finite2(B),B2)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B2)) ) ) ) ).

% card_inj_on_le
tff(fact_5821_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( ? [F7: fun(A,B)] :
              ( inj_on(A,B,F7,A4)
              & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A4)),B2) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B2)) ) ) ) ).

% inj_on_iff_card_le
tff(fact_5822_inj__on__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B2: set(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
    <=> ( inj_on(A,B,F2,A4)
        & inj_on(A,B,F2,B2)
        & ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),minus_minus(set(A),A4),B2))),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),minus_minus(set(A),B2),A4))) = bot_bot(set(B)) ) ) ) ).

% inj_on_Un
tff(fact_5823_log__inj,axiom,
    ! [B3: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),one_one(real)),B3)
     => inj_on(real,real,log(B3),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))) ) ).

% log_inj
tff(fact_5824_compact__imp__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [T4: set(A)] :
                ( member(set(A),T4,F3)
               => topolo7761053866217962861closed(A,T4) )
           => ( ! [F11: set(set(A))] :
                  ( aa(set(set(A)),$o,finite_finite2(set(A)),F11)
                 => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),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_5825_compact__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U2: set(A)] :
          ( topolo2193935891317330818ompact(A,U2)
        <=> ! [A8: set(set(A))] :
              ( ! [X: set(A)] :
                  ( member(set(A),X,A8)
                 => topolo7761053866217962861closed(A,X) )
             => ( ! [B11: set(set(A))] :
                    ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),B11),A8)
                   => ( aa(set(set(A)),$o,finite_finite2(set(A)),B11)
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B11)) != bot_bot(set(A)) ) ) )
               => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A8)) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_fip
tff(fact_5826_pos__deriv__imp__strict__mono,axiom,
    ! [F2: fun(real,real),F9: fun(real,real)] :
      ( ! [X3: real] : has_field_derivative(real,F2,aa(real,real,F9,X3),topolo174197925503356063within(real,X3,top_top(set(real))))
     => ( ! [X3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,F9,X3))
       => order_strict_mono(real,real,F2) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_5827_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),$o)] :
      ( ! [T9: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S))
         => aa(set(A),$o,P,T9) )
    <=> ! [T9: set(B)] :
          ( ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S)
            & inj_on(B,A,F2,T9) )
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% all_subset_image_inj
tff(fact_5828_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),$o)] :
      ( ? [T9: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S))
          & aa(set(A),$o,P,T9) )
    <=> ? [T9: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S)
          & inj_on(B,A,F2,T9)
          & aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% ex_subset_image_inj
tff(fact_5829_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C2: set(A),B2: set(A),Xa: A] :
      ( inj_on(A,B,G,C2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))
       => member(fun(B,A),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_yt(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C2),Xa),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)),B2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_5830_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_5831_inj__Suc,axiom,
    ! [N4: set(nat)] : inj_on(nat,nat,suc,N4) ).

% inj_Suc
tff(fact_5832_inj__singleton,axiom,
    ! [A: $tType,A4: set(A)] : inj_on(A,set(A),aTP_Lamp_lf(A,set(A)),A4) ).

% inj_singleton
tff(fact_5833_inj__on__diff__nat,axiom,
    ! [N4: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( member(nat,N2,N4)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2) )
     => inj_on(nat,nat,aTP_Lamp_jo(nat,fun(nat,nat),K),N4) ) ).

% inj_on_diff_nat
tff(fact_5834_inj__on__set__encode,axiom,
    inj_on(set(nat),nat,nat_set_encode,aa(fun(set(nat),$o),set(set(nat)),collect(set(nat)),finite_finite2(nat))) ).

% inj_on_set_encode
tff(fact_5835_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_yv(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_5836_range__inj__infinite,axiom,
    ! [A: $tType,F2: fun(nat,A)] :
      ( inj_on(nat,A,F2,top_top(set(nat)))
     => ~ aa(set(A),$o,finite_finite2(A),aa(set(nat),set(A),image2(nat,A,F2),top_top(set(nat)))) ) ).

% range_inj_infinite
tff(fact_5837_inj__enumerate,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),S)
         => inj_on(nat,A,infini527867602293511546merate(A,S),top_top(set(nat))) ) ) ).

% inj_enumerate
tff(fact_5838_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ? [N2: nat,F5: fun(nat,A)] :
          ( ( A4 = aa(set(nat),set(A),image2(nat,A,F5),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),N2))) )
          & inj_on(nat,A,F5,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_5839_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ? [F5: fun(A,nat),N2: nat] :
          ( ( aa(set(A),set(nat),image2(A,nat,F5),A4) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),N2)) )
          & inj_on(A,nat,F5,A4) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_5840_ge__eq__refl,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xa: A] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R)
     => aa(A,$o,aa(A,fun(A,$o),R,Xa),Xa) ) ).

% ge_eq_refl
tff(fact_5841_refl__ge__eq,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( ! [X3: A] : aa(A,$o,aa(A,fun(A,$o),R,X3),X3)
     => aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R) ) ).

% refl_ge_eq
tff(fact_5842_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X3: nat] :
            ( member(nat,X3,I5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs)) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_5843_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
    <=> ? [F7: fun(nat,A)] :
          ( inj_on(nat,A,F7,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F7),top_top(set(nat)))),S) ) ) ).

% infinite_iff_countable_subset
tff(fact_5844_infinite__countable__subset,axiom,
    ! [A: $tType,S: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),S)
     => ? [F5: fun(nat,A)] :
          ( inj_on(nat,A,F5,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,F5),top_top(set(nat)))),S) ) ) ).

% infinite_countable_subset
tff(fact_5845_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] : aa(real,$o,aa(real,fun(real,$o),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_5846_inj__on__funpow__least,axiom,
    ! [A: $tType,Na: nat,F2: fun(A,A),S3: A] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),F2),S3) = S3 )
     => ( ! [M4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M4)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M4),Na)
             => ( 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),S3) != S3 ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_yw(fun(A,A),fun(A,fun(nat,A)),F2),S3),set_or7035219750837199246ssThan(nat,zero_zero(nat),Na)) ) ) ).

% inj_on_funpow_least
tff(fact_5847_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] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X3))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F2),G))),suminf(real,F2)) ) ) ) ).

% suminf_reindex_mono
tff(fact_5848_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] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F2,X3))
         => ( ! [X3: nat] :
                ( ~ member(nat,X3,aa(set(nat),set(nat),image2(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_5849_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F13: fun(B,A),A15: set(B),B1: set(A),F24: fun(C,D),B22: set(C),A25: set(D)] :
      ( ( aa(set(B),set(A),image2(B,A,F13),A15) = B1 )
     => ( inj_on(C,D,F24,B22)
       => ( aa(set(D),$o,aa(set(D),fun(set(D),$o),ord_less_eq(set(D)),aa(set(C),set(D),image2(C,D,F24),B22)),A25)
         => ( ( ( B22 = bot_bot(set(C)) )
             => ( A25 = bot_bot(set(D)) ) )
           => ( bNF_Wellorder_Func(C,A,B22,B1) = aa(set(fun(D,B)),set(fun(C,A)),image2(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B22,F13,F24)),bNF_Wellorder_Func(D,B,A25,A15)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_5850_Func__non__emp,axiom,
    ! [A: $tType,B: $tType,B2: set(A),A4: set(B)] :
      ( ( B2 != bot_bot(set(A)) )
     => ( bNF_Wellorder_Func(B,A,A4,B2) != bot_bot(set(fun(B,A))) ) ) ).

% Func_non_emp
tff(fact_5851_Func__is__emp,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ( bNF_Wellorder_Func(A,B,A4,B2) = bot_bot(set(fun(A,B))) )
    <=> ( ( A4 != bot_bot(set(A)) )
        & ( B2 = bot_bot(set(B)) ) ) ) ).

% Func_is_emp
tff(fact_5852_Func__map,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,G: fun(A,B),A25: set(A),A15: set(B),F13: fun(B,C),B1: set(C),F24: fun(D,A),B22: set(D)] :
      ( member(fun(A,B),G,bNF_Wellorder_Func(A,B,A25,A15))
     => ( aa(set(C),$o,aa(set(C),fun(set(C),$o),ord_less_eq(set(C)),aa(set(B),set(C),image2(B,C,F13),A15)),B1)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(D),set(A),image2(D,A,F24),B22)),A25)
         => member(fun(D,C),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B22,F13,F24),G),bNF_Wellorder_Func(D,C,B22,B1)) ) ) ) ).

% Func_map
tff(fact_5853_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(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))),image2(nat,set(product_prod(A,A)),aTP_Lamp_xz(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_ye(set(product_prod(A,A)),fun(nat,$o),R)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_5854_has__derivative__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(A) )
     => ! [F2: fun(B,A),Xa: B,F9: fun(B,A),S: set(B),Na: int] :
          ( ( aa(B,A,F2,Xa) != zero_zero(A) )
         => ( has_derivative(B,A,F2,F9,topolo174197925503356063within(B,Xa,S))
           => has_derivative(B,A,aa(int,fun(B,A),aTP_Lamp_yx(fun(B,A),fun(int,fun(B,A)),F2),Na),aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_yy(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),F2),Xa),F9),Na),topolo174197925503356063within(B,Xa,S)) ) ) ) ).

% has_derivative_power_int
tff(fact_5855_has__derivative__power__int_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xa: A,Na: int,S: set(A)] :
          ( ( Xa != zero_zero(A) )
         => has_derivative(A,A,aTP_Lamp_yz(int,fun(A,A),Na),aa(int,fun(A,A),aTP_Lamp_za(A,fun(int,fun(A,A)),Xa),Na),topolo174197925503356063within(A,Xa,S)) ) ) ).

% has_derivative_power_int'
tff(fact_5856_power__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,Na: int] :
          ( ( power_int(A,Xa,Na) = zero_zero(A) )
        <=> ( ( Xa = zero_zero(A) )
            & ( Na != zero_zero(int) ) ) ) ) ).

% power_int_eq_0_iff
tff(fact_5857_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_5858_power__int__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,Na)),power_int(A,B3,Na))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ) ) ) ).

% power_int_mono_iff
tff(fact_5859_power__int__not__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,Na: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( Na = zero_zero(int) ) )
         => ( power_int(A,Xa,Na) != zero_zero(A) ) ) ) ).

% power_int_not_zero
tff(fact_5860_zero__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),power_int(A,Xa,Na)) ) ) ).

% zero_less_power_int
tff(fact_5861_rtrancl__subset__rtrancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),transitive_rtrancl(A,S3))
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R2)),transitive_rtrancl(A,S3)) ) ).

% rtrancl_subset_rtrancl
tff(fact_5862_rtrancl__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),transitive_rtrancl(A,R))
       => ( transitive_rtrancl(A,S) = transitive_rtrancl(A,R) ) ) ) ).

% rtrancl_subset
tff(fact_5863_rtrancl__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S3)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R2)),transitive_rtrancl(A,S3)) ) ).

% rtrancl_mono
tff(fact_5864_zero__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),power_int(A,Xa,Na)) ) ) ).

% zero_le_power_int
tff(fact_5865_rtrancl__Un__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),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_5866_continuous__on__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),Na: int] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( ! [X3: A] :
                ( member(A,X3,S3)
               => ( aa(A,B,F2,X3) != zero_zero(B) ) )
           => topolo81223032696312382ous_on(A,B,S3,aa(int,fun(A,B),aTP_Lamp_zb(fun(A,B),fun(int,fun(A,B)),F2),Na)) ) ) ) ).

% continuous_on_power_int
tff(fact_5867_power__int__0__left__If,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [M: int] :
          power_int(A,zero_zero(A),M) = $ite(M = zero_zero(int),one_one(A),zero_zero(A)) ) ).

% power_int_0_left_If
tff(fact_5868_power__int__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: int,N4: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,Na)),power_int(A,A3,N4)) ) ) ) ).

% power_int_increasing
tff(fact_5869_power__int__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: int,N4: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Na),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,Na)),power_int(A,A3,N4)) ) ) ) ).

% power_int_strict_increasing
tff(fact_5870_power__int__diff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,M: int,Na: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( M != Na ) )
         => ( power_int(A,Xa,aa(int,int,minus_minus(int,M),Na)) = divide_divide(A,power_int(A,Xa,M),power_int(A,Xa,Na)) ) ) ) ).

% power_int_diff
tff(fact_5871_tendsto__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F2: fun(A,B),A3: B,F3: filter(A),Na: int] :
          ( filterlim(A,B,F2,topolo7230453075368039082e_nhds(B,A3),F3)
         => ( ( A3 != zero_zero(B) )
           => filterlim(A,B,aa(int,fun(A,B),aTP_Lamp_zc(fun(A,B),fun(int,fun(A,B)),F2),Na),topolo7230453075368039082e_nhds(B,power_int(B,A3,Na)),F3) ) ) ) ).

% tendsto_power_int
tff(fact_5872_continuous__at__within__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [A3: A,S3: set(A),F2: fun(A,B),Na: int] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),F2)
         => ( ( aa(A,B,F2,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S3),aa(int,fun(A,B),aTP_Lamp_zd(fun(A,B),fun(int,fun(A,B)),F2),Na)) ) ) ) ).

% continuous_at_within_power_int
tff(fact_5873_differentiable__power__int,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F2: fun(A,B),Xa: A,S3: set(A),Na: int] :
          ( differentiable(A,B,F2,topolo174197925503356063within(A,Xa,S3))
         => ( ( aa(A,B,F2,Xa) != zero_zero(B) )
           => differentiable(A,B,aa(int,fun(A,B),aTP_Lamp_ze(fun(A,B),fun(int,fun(A,B)),F2),Na),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% differentiable_power_int
tff(fact_5874_continuous__power__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & topological_t2_space(A) )
     => ! [F3: filter(A),F2: fun(A,B),Na: int] :
          ( topolo3448309680560233919inuous(A,B,F3,F2)
         => ( ( aa(A,B,F2,topolo3827282254853284352ce_Lim(A,A,F3,aTP_Lamp_rz(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F3,aa(int,fun(A,B),aTP_Lamp_zd(fun(A,B),fun(int,fun(A,B)),F2),Na)) ) ) ) ).

% continuous_power_int
tff(fact_5875_power__int__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: int,N4: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Na),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,N4)),power_int(A,A3,Na)) ) ) ) ) ).

% power_int_strict_decreasing
tff(fact_5876_power__int__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xa,Na)),power_int(A,Ya,Na)) ) ) ) ) ).

% power_int_mono
tff(fact_5877_power__int__strict__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Na),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,B3,Na)),power_int(A,A3,Na)) ) ) ) ) ).

% power_int_strict_antimono
tff(fact_5878_one__le__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xa)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),power_int(A,Xa,Na)) ) ) ) ).

% one_le_power_int
tff(fact_5879_one__less__power__int,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),A3)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
           => aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),power_int(A,A3,Na)) ) ) ) ).

% one_less_power_int
tff(fact_5880_power__int__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: int,Na: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( aa(int,int,aa(int,fun(int,int),plus_plus(int),M),Na) != zero_zero(int) ) )
         => ( power_int(A,Xa,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),Na)) = aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,M)),power_int(A,Xa,Na)) ) ) ) ).

% power_int_add
tff(fact_5881_power__int__antimono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),A3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Na),zero_zero(int))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,B3,Na)),power_int(A,A3,Na)) ) ) ) ) ).

% power_int_antimono
tff(fact_5882_power__int__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na)
             => aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,A3,Na)),power_int(A,B3,Na)) ) ) ) ) ).

% power_int_strict_mono
tff(fact_5883_power__int__le__one,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xa)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xa,Na)),one_one(A)) ) ) ) ) ).

% power_int_le_one
tff(fact_5884_power__int__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Na: int,N4: int,A3: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Na),N4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),one_one(A))
             => ( ( ( A3 != zero_zero(A) )
                  | ( N4 != zero_zero(int) )
                  | ( Na = zero_zero(int) ) )
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,A3,N4)),power_int(A,A3,Na)) ) ) ) ) ) ).

% power_int_decreasing
tff(fact_5885_power__int__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,M: int,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),power_int(A,Xa,M)),power_int(A,Xa,Na))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),Na) ) ) ) ) ).

% power_int_le_imp_le_exp
tff(fact_5886_power__int__le__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,M: int,Na: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),Xa)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),power_int(A,Xa,M)),power_int(A,Xa,Na))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na)
             => aa(int,$o,aa(int,fun(int,$o),ord_less(int),M),Na) ) ) ) ) ).

% power_int_le_imp_less_exp
tff(fact_5887_power__int__minus__mult,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xa: A,Na: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( Na != zero_zero(int) ) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),power_int(A,Xa,aa(int,int,minus_minus(int,Na),one_one(int)))),Xa) = power_int(A,Xa,Na) ) ) ) ).

% power_int_minus_mult
tff(fact_5888_power__int__add__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xa,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,Xa,M)),Xa) ) ) ) ).

% power_int_add_1
tff(fact_5889_power__int__add__1_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: A,M: int] :
          ( ( ( Xa != zero_zero(A) )
            | ( M != aa(int,int,uminus_uminus(int),one_one(int)) ) )
         => ( power_int(A,Xa,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),Xa),power_int(A,Xa,M)) ) ) ) ).

% power_int_add_1'
tff(fact_5890_power__int__def,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & power(A) )
     => ! [Xa: A,Na: int] :
          power_int(A,Xa,Na) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Na),aa(nat,A,power_power(A,Xa),aa(int,nat,nat2,Na)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),Xa)),aa(int,nat,nat2,aa(int,int,uminus_uminus(int),Na)))) ) ).

% power_int_def
tff(fact_5891_powr__real__of__int_H,axiom,
    ! [Xa: real,Na: int] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( ( ( Xa != zero_zero(real) )
          | aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Na) )
       => ( powr(real,Xa,aa(int,real,ring_1_of_int(real),Na)) = power_int(real,Xa,Na) ) ) ) ).

% powr_real_of_int'
tff(fact_5892_DERIV__power__int,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F2: fun(A,A),D2: A,Xa: A,S3: set(A),Na: int] :
          ( has_field_derivative(A,F2,D2,topolo174197925503356063within(A,Xa,S3))
         => ( ( aa(A,A,F2,Xa) != zero_zero(A) )
           => has_field_derivative(A,aa(int,fun(A,A),aTP_Lamp_zf(fun(A,A),fun(int,fun(A,A)),F2),Na),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),Na)),power_int(A,aa(A,A,F2,Xa),aa(int,int,minus_minus(int,Na),one_one(int))))),D2),topolo174197925503356063within(A,Xa,S3)) ) ) ) ).

% DERIV_power_int
tff(fact_5893_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2),lex(A,R2))
     => ~ ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Ys2))
             => ( ( take(A,I2,Xs) = take(A,I2,Ys2) )
               => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Ys2),I2)),R2) ) ) ) ) ).

% lex_take_index
tff(fact_5894_prod__filter__def,axiom,
    ! [A: $tType,B: $tType,F3: filter(A),G2: filter(B)] : prod_filter(A,B,F3,G2) = aa(set(filter(product_prod(A,B))),filter(product_prod(A,B)),complete_Inf_Inf(filter(product_prod(A,B))),aa(set(product_prod(fun(A,$o),fun(B,$o))),set(filter(product_prod(A,B))),image2(product_prod(fun(A,$o),fun(B,$o)),filter(product_prod(A,B)),aa(fun(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),fun(product_prod(fun(A,$o),fun(B,$o)),filter(product_prod(A,B))),product_case_prod(fun(A,$o),fun(B,$o),filter(product_prod(A,B))),aTP_Lamp_zh(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))))),aa(fun(product_prod(fun(A,$o),fun(B,$o)),$o),set(product_prod(fun(A,$o),fun(B,$o))),collect(product_prod(fun(A,$o),fun(B,$o))),aa(fun(fun(A,$o),fun(fun(B,$o),$o)),fun(product_prod(fun(A,$o),fun(B,$o)),$o),product_case_prod(fun(A,$o),fun(B,$o),$o),aa(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o)),aTP_Lamp_zi(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),F3),G2))))) ).

% prod_filter_def
tff(fact_5895_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xa,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n2: nat,
                    n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                    $let(
                      m: nat,
                      m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                      ( ! [X: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                         => vEBT_VEBT_valid(X,n2) )
                      & vEBT_VEBT_valid(Summary3,m)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X8)
                          & ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X8) ) ),
                          aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5896_ball__empty,axiom,
    ! [A: $tType,P: fun(A,$o),X2: A] :
      ( member(A,X2,bot_bot(set(A)))
     => aa(A,$o,P,X2) ) ).

% ball_empty
tff(fact_5897_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zk(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),P),Q)))
      <=> ! [Y4: A] :
            ( aa(A,$o,P,Y4)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),Y4))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_5898_ball__UNIV,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,top_top(set(A)))
         => aa(A,$o,P,X) )
    <=> ! [X_13: A] : aa(A,$o,P,X_13) ) ).

% ball_UNIV
tff(fact_5899_Ball__Collect,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,A4)
         => aa(A,$o,P,X) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(fun(A,$o),set(A),collect(A),P)) ) ).

% Ball_Collect
tff(fact_5900_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_zm(fun(A,fun(B,$o)),fun(A,$o),P)),F3)
    <=> ? [Y5: fun(A,B)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_zn(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y5)),F3) ) ).

% eventually_ex
tff(fact_5901_Ball__fold,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X: A] :
            ( member(A,X,A4)
           => aa(A,$o,P,X) )
      <=> finite_fold(A,$o,aTP_Lamp_zo(fun(A,$o),fun(A,fun($o,$o)),P),$true,A4) ) ) ).

% Ball_fold
tff(fact_5902_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F2: fun(A,B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(A,B),fun(B,$o),aTP_Lamp_zp(fun(A,$o),fun(fun(A,B),fun(B,$o)),P),F2))) ) ).

% finite_image_set
tff(fact_5903_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),F2: fun(A,fun(B,C))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),Q))
       => aa(set(C),$o,finite_finite2(C),aa(fun(C,$o),set(C),collect(C),aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_zq(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P),Q),F2))) ) ) ).

% finite_image_set2
tff(fact_5904_finite_Omono,axiom,
    ! [A: $tType] : aa(fun(fun(set(A),$o),fun(set(A),$o)),$o,order_mono(fun(set(A),$o),fun(set(A),$o)),aTP_Lamp_zr(fun(set(A),$o),fun(set(A),$o))) ).

% finite.mono
tff(fact_5905_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] : aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_zs(fun(B,A),fun(set(B),fun(A,$o)),F2),A4)) = aa(set(B),set(A),image2(B,A,F2),A4) ).

% Setcompr_eq_image
tff(fact_5906_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),P: fun(B,$o)] : aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_zt(fun(B,A),fun(fun(B,$o),fun(A,$o)),F2),P)) = aa(set(B),set(A),image2(B,A,F2),aa(fun(B,$o),set(B),collect(B),P)) ).

% setcompr_eq_image
tff(fact_5907_closed__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_zu(product_prod(A,A),$o))) ) ).

% closed_superdiagonal
tff(fact_5908_closed__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_zv(product_prod(A,A),$o))) ) ).

% closed_subdiagonal
tff(fact_5909_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_zw(product_prod(A,A),$o))) ) ).

% open_superdiagonal
tff(fact_5910_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_zx(product_prod(A,A),$o))) ) ).

% open_subdiagonal
tff(fact_5911_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A)] : aa(fun(A,$o),set(A),collect(A),aTP_Lamp_zy(fun(B,A),fun(A,$o),F2)) = aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_5912_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X3: A] :
            ( member(A,X3,A4)
           => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X3)),Net) )
       => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zz(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P)),Net) ) ) ).

% eventually_ball_finite
tff(fact_5913_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zz(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P)),Net)
      <=> ! [X: A] :
            ( member(A,X,A4)
           => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X)),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_5914_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A4) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aaa(set(A),fun(A,$o),A4))) ) ).

% Sup_eq_Inf
tff(fact_5915_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A4) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aab(set(A),fun(A,$o),A4))) ) ).

% Inf_eq_Sup
tff(fact_5916_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aac(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_5917_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),Xa),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aad(set(A),fun(A,fun(A,$o)),A4),Xa))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_5918_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A4)),aa(set(A),A,lattic5882676163264333800up_fin(A),B2)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aae(set(A),fun(set(A),fun(A,$o)),A4),B2))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_5919_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),Xa),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aaf(set(A),fun(A,fun(A,$o)),A4),Xa))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_5920_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A4)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B2)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aag(set(A),fun(set(A),fun(A,$o)),A4),B2))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_5921_cInf__cSup,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( condit1013018076250108175_below(A,S)
           => ( aa(set(A),A,complete_Inf_Inf(A),S) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aah(set(A),fun(A,$o),S))) ) ) ) ) ).

% cInf_cSup
tff(fact_5922_cSup__cInf,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [S: set(A)] :
          ( ( S != bot_bot(set(A)) )
         => ( condit941137186595557371_above(A,S)
           => ( aa(set(A),A,complete_Sup_Sup(A),S) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aai(set(A),fun(A,$o),S))) ) ) ) ) ).

% cSup_cInf
tff(fact_5923_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 )
        & $let(
            n2: nat,
            n2:= divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),bit0(one2))),
            $let(
              m: nat,
              m:= aa(nat,nat,minus_minus(nat,Deg),n2),
              ( ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                 => vEBT_VEBT_valid(X,n2) )
              & vEBT_VEBT_valid(Summary,m)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X8)
                  & ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X8) ) ),
                  aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList),Summary),n2),m)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5924_funpow__inj__finite,axiom,
    ! [A: $tType,P3: fun(A,A),Xa: A] :
      ( inj_on(A,A,P3,top_top(set(A)))
     => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aaj(fun(A,A),fun(A,fun(A,$o)),P3),Xa)))
       => ~ ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),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),Xa) != Xa ) ) ) ) ).

% funpow_inj_finite
tff(fact_5925_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( vEBT_VEBT_valid(Xa,Xaa)
      <=> (Ya) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( (Ya)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ( (Ya)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => vEBT_VEBT_valid(X,n2) )
                          & vEBT_VEBT_valid(Summary3,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X8)
                              & ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X8) ) ),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5926_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xa,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xa = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n2: nat,
                      n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $let(
                        m: nat,
                        m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                        ( ! [X: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                           => vEBT_VEBT_valid(X,n2) )
                        & vEBT_VEBT_valid(Summary3,m)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X8)
                            & ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X8) ) ),
                            aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5927_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat,Ya: $o] :
      ( ( vEBT_VEBT_valid(Xa,Xaa)
      <=> (Ya) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Ya)
                <=> ( Xaa = one_one(nat) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,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))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( ( (Ya)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => vEBT_VEBT_valid(X,n2) )
                            & vEBT_VEBT_valid(Summary3,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X8)
                                & ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X8) ) ),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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,Summary3)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_5928_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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))),Xaa))
               => ( Xaa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Summary3)),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                               => vEBT_VEBT_valid(X,n2) )
                            & vEBT_VEBT_valid(Summary3,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X8)
                                & ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                   => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X8) ) ),
                                aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5929_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xa: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xa,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,Xa),Xaa))
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( Xa = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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))),Xaa))
               => ( Xaa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Mima,Deg2,TreeList2,Summary3) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Summary3)),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                             => vEBT_VEBT_valid(X,n2) )
                          & vEBT_VEBT_valid(Summary3,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X8)
                              & ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                                 => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X8) ) ),
                              aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList2),Summary3),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5930_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aak(set(set(A)),fun(set(A),$o),A4))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A4))) ) ).

% Sup_Inf_le
tff(fact_5931_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A4: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aal(set(set(A)),fun(set(A),$o),A4))))) ) ).

% Inf_Sup_le
tff(fact_5932_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A4: set(set(A))] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Sup_Sup(A)),A4))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image2(set(A),A,complete_Inf_Inf(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aam(set(set(A)),fun(set(A),$o),A4))))) ) ).

% finite_Inf_Sup
tff(fact_5933_mono__compose,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Q: fun(A,fun(B,C)),F2: fun(D,B)] :
          ( aa(fun(A,fun(B,C)),$o,order_mono(A,fun(B,C)),Q)
         => aa(fun(A,fun(D,C)),$o,order_mono(A,fun(D,C)),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aan(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Q),F2)) ) ) ).

% mono_compose
tff(fact_5934_Pow__Compl,axiom,
    ! [A: $tType,A4: set(A)] : pow2(A,aa(set(A),set(A),uminus_uminus(set(A)),A4)) = aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aao(set(A),fun(set(A),$o),A4)) ).

% Pow_Compl
tff(fact_5935_Sup__real__def,axiom,
    ! [X4: set(real)] : aa(set(real),real,complete_Sup_Sup(real),X4) = ord_Least(real,aTP_Lamp_aap(set(real),fun(real,$o),X4)) ).

% Sup_real_def
tff(fact_5936_Sup__int__def,axiom,
    ! [X4: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X4) = the(int,aTP_Lamp_aaq(set(int),fun(int,$o),X4)) ).

% Sup_int_def
tff(fact_5937_Union__maximal__sets,axiom,
    ! [A: $tType,F14: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),F14)
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aar(set(set(A)),fun(set(A),$o),F14))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F14) ) ) ).

% Union_maximal_sets
tff(fact_5938_Inf__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(fun(filter(A),$o),set(filter(A)),collect(filter(A)),aTP_Lamp_aas(set(filter(A)),fun(filter(A),$o),S))) ).

% Inf_filter_def
tff(fact_5939_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A)] : aa(fun(fun(A,$o),fun(A,$o)),$o,order_mono(fun(A,$o),fun(A,$o)),aTP_Lamp_aat(fun(A,A),fun(fun(A,$o),fun(A,$o)),F2)) ) ).

% iteratesp.mono
tff(fact_5940_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2),listrel1(A,R2))
    <=> ? [Y4: A,N: nat] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),N)),Y4),R2)
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
          & ( Ys2 = list_update(A,Xs,N,Y4) ) ) ) ).

% listrel1_iff_update
tff(fact_5941_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = aa(fun(product_prod(list(A),list(A)),$o),set(product_prod(list(A),list(A))),collect(product_prod(list(A),list(A))),aa(fun(list(A),fun(list(A),$o)),fun(product_prod(list(A),list(A)),$o),product_case_prod(list(A),list(A),$o),aTP_Lamp_aau(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lenlex_conv
tff(fact_5942_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,transitive_rtrancl(A,R2))),transitive_rtrancl(list(A),listrel1(A,R2))) ).

% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_5943_listrel1__mono,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S3)
     => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel1(A,S3)) ) ).

% listrel1_mono
tff(fact_5944_chain__subset,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o)),A4: set(A),B2: set(A)] :
      ( comple1602240252501008431_chain(A,Ord,A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
       => comple1602240252501008431_chain(A,Ord,B2) ) ) ).

% chain_subset
tff(fact_5945_chain__empty,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o))] : comple1602240252501008431_chain(A,Ord,bot_bot(set(A))) ).

% chain_empty
tff(fact_5946_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A4: set(A),Z: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),Z) ) ) ) ).

% ccpo_Sup_least
tff(fact_5947_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A4: set(A),Xa: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
         => ( member(A,Xa,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ).

% ccpo_Sup_upper
tff(fact_5948_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Xa: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_5949_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ms),Ns),lenlex(A,R2))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_5950_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A4: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A4)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => member(A,aa(set(A),A,complete_Sup_Sup(A),A4),A4) ) ) ) ) ).

% in_chain_finite
tff(fact_5951_set__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),set(A),set2(A),nths(A,Xs,I5)) = aa(fun(A,$o),set(A),collect(A),aa(set(nat),fun(A,$o),aTP_Lamp_aav(list(A),fun(set(nat),fun(A,$o)),Xs),I5)) ).

% set_nths
tff(fact_5952_finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( finite5375528669736107172at_top(A,A4) = principal(set(A),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),A4),bot_bot(set(set(A))))) ) ) ).

% finite_subsets_at_top_finite
tff(fact_5953_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),$o)] :
      ( ! [X6: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X6)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X6),A4)
           => aa(set(A),$o,P,X6) ) )
     => aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),P),finite5375528669736107172at_top(A,A4)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_5954_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_5955_eventually__finite__subsets__at__top__finite,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),P),finite5375528669736107172at_top(A,A4))
      <=> aa(set(A),$o,P,A4) ) ) ).

% eventually_finite_subsets_at_top_finite
tff(fact_5956_finite__subsets__at__top__neq__bot,axiom,
    ! [A: $tType,A4: set(A)] : finite5375528669736107172at_top(A,A4) != bot_bot(filter(set(A))) ).

% finite_subsets_at_top_neq_bot
tff(fact_5957_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),nths(A,Xs,I5))),aa(list(A),set(A),set2(A),Xs)) ).

% set_nths_subset
tff(fact_5958_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
         => member(nat,I2,I5) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_5959_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,P: fun(set(A),$o),A4: set(A)] :
      ( aa(filter(set(A)),$o,aa(fun(set(A),$o),fun(filter(set(A)),$o),eventually(set(A)),P),finite5375528669736107172at_top(A,A4))
    <=> ? [X8: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X8)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X8),A4)
          & ! [Y5: set(A)] :
              ( ( aa(set(A),$o,finite_finite2(A),Y5)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X8),Y5)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Y5),A4) )
             => aa(set(A),$o,P,Y5) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_5960_length__nths,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : aa(list(A),nat,size_size(list(A)),nths(A,Xs,I5)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_aaw(list(A),fun(set(nat),fun(nat,$o)),Xs),I5))) ).

% length_nths
tff(fact_5961_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A4: set(A)] : finite5375528669736107172at_top(A,A4) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image2(set(A),filter(set(A)),aTP_Lamp_aay(set(A),fun(set(A),filter(set(A))),A4)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aaz(set(A),fun(set(A),$o),A4)))) ).

% finite_subsets_at_top_def
tff(fact_5962_filterlim__lessThan__at__top,axiom,
    filterlim(nat,set(nat),set_ord_lessThan(nat),finite5375528669736107172at_top(nat,top_top(set(nat))),at_top(nat)) ).

% filterlim_lessThan_at_top
tff(fact_5963_filterlim__atMost__at__top,axiom,
    filterlim(nat,set(nat),set_ord_atMost(nat),finite5375528669736107172at_top(nat,top_top(set(nat))),at_top(nat)) ).

% filterlim_atMost_at_top
tff(fact_5964_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,set(B)),A4: set(B),F3: filter(A)] :
      ( filterlim(A,set(B),F2,finite5375528669736107172at_top(B,A4),F3)
    <=> ! [X8: set(B)] :
          ( ( aa(set(B),$o,finite_finite2(B),X8)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),X8),A4) )
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_aba(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),F2),A4),X8)),F3) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_5965_flat__lub__def,axiom,
    ! [A: $tType,B3: A,A4: set(A)] :
      partial_flat_lub(A,B3,A4) = $ite(aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))),B3,the(A,aa(set(A),fun(A,$o),aTP_Lamp_abb(A,fun(set(A),fun(A,$o)),B3),A4))) ).

% flat_lub_def
tff(fact_5966_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = aa(fun(real,$o),set(real),collect(real),aTP_Lamp_abc(real,$o)) ).

% Rats_eq_int_div_nat
tff(fact_5967_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A4) = the2(A,finite_fold(A,option(A),aTP_Lamp_abd(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Max.eq_fold'
tff(fact_5968_Rats__no__top__le,axiom,
    ! [Xa: real] :
    ? [X3: real] :
      ( member(real,X3,field_char_0_Rats(real))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),X3) ) ).

% Rats_no_top_le
tff(fact_5969_Rats__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => member(A,zero_zero(A),field_char_0_Rats(A)) ) ).

% Rats_0
tff(fact_5970_Rats__no__bot__less,axiom,
    ! [Xa: real] :
    ? [X3: real] :
      ( member(real,X3,field_char_0_Rats(real))
      & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),Xa) ) ).

% Rats_no_bot_less
tff(fact_5971_Rats__dense__in__real,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
     => ? [X3: real] :
          ( member(real,X3,field_char_0_Rats(real))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),X3)
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),X3),Ya) ) ) ).

% Rats_dense_in_real
tff(fact_5972_Rats__infinite,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ~ aa(set(A),$o,finite_finite2(A),field_char_0_Rats(A)) ) ).

% Rats_infinite
tff(fact_5973_Ints__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),ring_1_Ints(A)),field_char_0_Rats(A)) ) ).

% Ints_subset_Rats
tff(fact_5974_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A4) = the2(A,none(A)) ) ) ) ).

% Max.infinite
tff(fact_5975_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = the2(A,none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_5976_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A4) = the2(A,none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_5977_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A4) = the2(A,finite_fold(A,option(A),aTP_Lamp_abe(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Inf_fin.eq_fold'
tff(fact_5978_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A4) = the2(A,finite_fold(A,option(A),aTP_Lamp_abf(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Sup_fin.eq_fold'
tff(fact_5979_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Na: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),Na)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Na)) ) ).

% take_bit_numeral_numeral
tff(fact_5980_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_abg(A,$o)) ) ) ).

% Nats_altdef1
tff(fact_5981_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_5982_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,one_one(A),semiring_1_Nats(A)) ) ).

% Nats_1
tff(fact_5983_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,semiring_1_Nats(A))
         => ( member(A,B3,semiring_1_Nats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),semiring_1_Nats(A)) ) ) ) ).

% Nats_add
tff(fact_5984_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => member(A,zero_zero(A),semiring_1_Nats(A)) ) ).

% Nats_0
tff(fact_5985_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,semiring_1_Nats(A))
         => ( member(A,B3,semiring_1_Nats(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3),semiring_1_Nats(A)) ) ) ) ).

% Nats_mult
tff(fact_5986_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] : member(A,aa(nat,A,semiring_1_of_nat(A),Na),semiring_1_Nats(A)) ) ).

% of_nat_in_Nats
tff(fact_5987_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xa: A,P: fun(A,$o)] :
          ( member(A,Xa,semiring_1_Nats(A))
         => ( ! [N2: nat] : aa(A,$o,P,aa(nat,A,semiring_1_of_nat(A),N2))
           => aa(A,$o,P,Xa) ) ) ) ).

% Nats_induct
tff(fact_5988_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xa: A] :
          ( member(A,Xa,semiring_1_Nats(A))
         => ~ ! [N2: nat] : Xa != aa(nat,A,semiring_1_of_nat(A),N2) ) ) ).

% Nats_cases
tff(fact_5989_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,semiring_1_Nats(A))
         => ( member(A,B3,semiring_1_Nats(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
             => member(A,aa(A,A,minus_minus(A,A3),B3),semiring_1_Nats(A)) ) ) ) ) ).

% Nats_diff
tff(fact_5990_Nats__subset__Ints,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),semiring_1_Nats(A)),ring_1_Ints(A)) ) ).

% Nats_subset_Ints
tff(fact_5991_Nats__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),semiring_1_Nats(A)),field_char_0_Rats(A)) ) ).

% Nats_subset_Rats
tff(fact_5992_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: num] :
          ( ( bit_take_bit_num(M,Na) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),Na)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5993_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = aa(set(nat),set(A),image2(nat,A,semiring_1_of_nat(A)),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_5994_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_abh(A,$o)) ) ) ).

% Nats_altdef2
tff(fact_5995_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(A,fun(B,B),F2,Xa),finite_fold(A,B,F2,Z,A4)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_5996_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( finite_fold(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = finite_fold(A,B,F2,aa(B,B,aa(A,fun(B,B),F2,Xa),Z),A4) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_5997_comp__fun__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => finite4664212375090638736ute_on(A,B,S,F2) ) ).

% comp_fun_idem_on.axioms(1)
tff(fact_5998_comp__fun__idem__on_Ofun__left__idem,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Z: B] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( member(A,Xa,S)
       => ( aa(B,B,aa(A,fun(B,B),F2,Xa),aa(B,B,aa(A,fun(B,B),F2,Xa),Z)) = aa(B,B,aa(A,fun(B,B),F2,Xa),Z) ) ) ) ).

% comp_fun_idem_on.fun_left_idem
tff(fact_5999_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( member(A,Xa,S)
       => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xa)),aa(A,fun(B,B),F2,Xa)) = aa(A,fun(B,B),F2,Xa) ) ) ) ).

% comp_fun_idem_on.comp_fun_idem_on
tff(fact_6000_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S: set(A),F2: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(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_6001_take__bit__num__def,axiom,
    ! [Na: nat,M: num] :
      bit_take_bit_num(Na,M) = $ite(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat),none(num),aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Na),aa(num,nat,numeral_numeral(nat),M))))) ).

% take_bit_num_def
tff(fact_6002_Sup__filter__def,axiom,
    ! [A: $tType,S: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),S) = abs_filter(A,aTP_Lamp_abi(set(filter(A)),fun(fun(A,$o),$o),S)) ).

% Sup_filter_def
tff(fact_6003_num__of__nat__numeral__eq,axiom,
    ! [Q5: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q5)) = Q5 ).

% num_of_nat_numeral_eq
tff(fact_6004_bot__filter__def,axiom,
    ! [A: $tType] : bot_bot(filter(A)) = abs_filter(A,aTP_Lamp_abj(fun(A,$o),$o)) ).

% bot_filter_def
tff(fact_6005_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_6006_sup__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),F3),F8) = abs_filter(A,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_abk(filter(A),fun(filter(A),fun(fun(A,$o),$o)),F3),F8)) ).

% sup_filter_def
tff(fact_6007_principal__def,axiom,
    ! [A: $tType,S: set(A)] : principal(A,S) = abs_filter(A,ball(A,S)) ).

% principal_def
tff(fact_6008_numeral__num__of__nat,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(Na)) = Na ) ) ).

% numeral_num_of_nat
tff(fact_6009_num__of__nat__One,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),one_one(nat))
     => ( num_of_nat(Na) = one2 ) ) ).

% num_of_nat_One
tff(fact_6010_map__filter__on__def,axiom,
    ! [A: $tType,B: $tType,X4: set(B),F2: fun(B,A),F3: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),map_filter_on(B,A,X4),F2),F3) = abs_filter(A,aa(filter(B),fun(fun(A,$o),$o),aa(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_abm(set(B),fun(fun(B,A),fun(filter(B),fun(fun(A,$o),$o))),X4),F2),F3)) ).

% map_filter_on_def
tff(fact_6011_top__filter__def,axiom,
    ! [A: $tType] : top_top(filter(A)) = abs_filter(A,fAll(A)) ).

% top_filter_def
tff(fact_6012_filtercomap__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F3: filter(B)] : filtercomap(A,B,F2,F3) = abs_filter(A,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_abn(fun(A,B),fun(filter(B),fun(fun(A,$o),$o)),F2),F3)) ).

% filtercomap_def
tff(fact_6013_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          aa(num,A,numeral_numeral(A),num_of_nat(Na)) = $ite(Na = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),Na)) ) ).

% numeral_num_of_nat_unfold
tff(fact_6014_inf__filter__def,axiom,
    ! [A: $tType,F3: filter(A),F8: filter(A)] : aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F3),F8) = abs_filter(A,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_abo(filter(A),fun(filter(A),fun(fun(A,$o),$o)),F3),F8)) ).

% inf_filter_def
tff(fact_6015_num__of__nat__double,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),Na)) = bit0(num_of_nat(Na)) ) ) ).

% num_of_nat_double
tff(fact_6016_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(Na)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_6017_num__of__nat_Osimps_I2_J,axiom,
    ! [Na: nat] :
      num_of_nat(aa(nat,nat,suc,Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na),inc(num_of_nat(Na)),one2) ).

% num_of_nat.simps(2)
tff(fact_6018_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_ta(A,fun(A,$o))) = ord_max(A) ) ) ).

% dual_min
tff(fact_6019_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: 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),Na))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) ) ).

% add_neg_numeral_special(5)
tff(fact_6020_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_6021_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Na)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Na))) ) ).

% diff_numeral_special(5)
tff(fact_6022_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(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_6023_num__induct,axiom,
    ! [P: fun(num,$o),Xa: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X3: num] :
            ( aa(num,$o,P,X3)
           => aa(num,$o,P,inc(X3)) )
       => aa(num,$o,P,Xa) ) ) ).

% num_induct
tff(fact_6024_ord_Omin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : min(A,Less_eq) = min(A,Less_eq) ).

% ord.min.cong
tff(fact_6025_ord_Omin__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),A3: A,B3: A] :
      aa(A,A,aa(A,fun(A,A),min(A,Less_eq),A3),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3),A3,B3) ).

% ord.min_def
tff(fact_6026_add__inc,axiom,
    ! [Xa: num,Ya: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),inc(Ya)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),Ya)) ).

% add_inc
tff(fact_6027_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_6028_inc_Osimps_I3_J,axiom,
    ! [Xa: num] : inc(bit1(Xa)) = bit0(inc(Xa)) ).

% inc.simps(3)
tff(fact_6029_inc_Osimps_I2_J,axiom,
    ! [Xa: num] : inc(bit0(Xa)) = bit1(Xa) ).

% inc.simps(2)
tff(fact_6030_add__One,axiom,
    ! [Xa: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xa),one2) = inc(Xa) ).

% add_One
tff(fact_6031_inc__BitM__eq,axiom,
    ! [Na: num] : inc(bitM(Na)) = bit0(Na) ).

% inc_BitM_eq
tff(fact_6032_BitM__inc__eq,axiom,
    ! [Na: num] : bitM(inc(Na)) = bit1(Na) ).

% BitM_inc_eq
tff(fact_6033_mult__inc,axiom,
    ! [Xa: num,Ya: num] : aa(num,num,aa(num,fun(num,num),times_times(num),Xa),inc(Ya)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),Xa),Ya)),Xa) ).

% mult_inc
tff(fact_6034_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xa: num] : aa(num,A,numeral_numeral(A),inc(Xa)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xa)),one_one(A)) ) ).

% numeral_inc
tff(fact_6035_sub__inc__One__eq,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Na: num] : neg_numeral_sub(A,inc(Na),one2) = aa(num,A,numeral_numeral(A),Na) ) ).

% sub_inc_One_eq
tff(fact_6036_card__Min__le__sum,axiom,
    ! [A: $tType,A4: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image2(A,nat,F2),A4)))),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),F2),A4)) ) ).

% card_Min_le_sum
tff(fact_6037_total__on__singleton,axiom,
    ! [A: $tType,Xa: A] : total_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Xa)),bot_bot(set(product_prod(A,A))))) ).

% total_on_singleton
tff(fact_6038_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ) ).

% Min_singleton
tff(fact_6039_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_6040_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_6041_Min__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A4: set(A),C3: B] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,aTP_Lamp_wt(B,fun(A,B),C3)),A4)) = C3 ) ) ) ) ).

% Min_const
tff(fact_6042_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798350308766er_Min(A),S)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_6043_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798349783984er_Max(A),S)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,uminus_uminus(A)),S)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_6044_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A4),A4) ) ) ) ).

% Min_in
tff(fact_6045_total__on__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : total_on(A,bot_bot(set(A)),R2) ).

% total_on_empty
tff(fact_6046_total__onI,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,A4)
         => ( member(A,Y,A4)
           => ( ( X3 != Y )
             => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y),R2)
                | member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),X3),R2) ) ) ) )
     => total_on(A,A4,R2) ) ).

% total_onI
tff(fact_6047_total__on__def,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A4,R2)
    <=> ! [X: A] :
          ( member(A,X,A4)
         => ! [Xa3: A] :
              ( member(A,Xa3,A4)
             => ( ( X != Xa3 )
               => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Xa3),R2)
                  | member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X),R2) ) ) ) ) ) ).

% total_on_def
tff(fact_6048_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_6049_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,A3,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),A3) ) ) ) ).

% Min.coboundedI
tff(fact_6050_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [Y: A] :
                ( member(A,Y,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y) )
           => ( member(A,Xa,A4)
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = Xa ) ) ) ) ) ).

% Min_eqI
tff(fact_6051_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),Xa) ) ) ) ).

% Min_le
tff(fact_6052_Least__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P))
         => ( ? [X_12: A] : aa(A,$o,P,X_12)
           => ( ord_Least(A,P) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),P)) ) ) ) ) ).

% Least_Min
tff(fact_6053_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = M )
            <=> ( member(A,M,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_6054_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),Xa)
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa) ) ) ) ) ) ).

% Min_le_iff
tff(fact_6055_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A4) )
            <=> ( member(A,M,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),X) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_6056_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A4))
             => ! [A10: A] :
                  ( member(A,A10,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A10) ) ) ) ) ) ).

% Min.boundedE
tff(fact_6057_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),A5) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ).

% Min.boundedI
tff(fact_6058_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),Xa)
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Xa) ) ) ) ) ) ).

% Min_less_iff
tff(fact_6059_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [B5: A] :
                ( member(A,B5,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B5) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = A3 ) ) ) ) ).

% Min_insert2
tff(fact_6060_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = aa(set(A),A,complete_Inf_Inf(A),A4) ) ) ) ) ).

% Min_Inf
tff(fact_6061_cInf__eq__Min,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),X4)
         => ( ( X4 != bot_bot(set(A)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X4) = aa(set(A),A,lattic643756798350308766er_Min(A),X4) ) ) ) ) ).

% cInf_eq_Min
tff(fact_6062_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( ~ aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = the2(A,none(A)) ) ) ) ).

% Min.infinite
tff(fact_6063_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M5: set(A),N4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M5),N4)
         => ( ( M5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),N4)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N4)),aa(set(A),A,lattic643756798350308766er_Min(A),M5)) ) ) ) ) ).

% Min_antimono
tff(fact_6064_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B2)),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ).

% Min.subset_imp
tff(fact_6065_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(set(A),$o,finite_finite2(A),A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(A,B,F2,aa(set(A),A,lattic643756798350308766er_Min(A),A4)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F2),A4)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_6066_Min__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S: set(A),F2: fun(A,B),K: B] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ( aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_wx(fun(A,B),fun(B,fun(A,B)),F2),K)),S)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,F2),S))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_6067_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_ta(A,fun(A,$o))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_6068_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A4) = the2(A,finite_fold(A,option(A),aTP_Lamp_abp(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Min.eq_fold'
tff(fact_6069_min__Suc__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)) ).

% min_Suc_Suc
tff(fact_6070_min__0R,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Na),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_6071_min__0L,axiom,
    ! [Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),Na) = zero_zero(nat) ).

% min_0L
tff(fact_6072_min_Oright__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) ) ).

% min.right_idem
tff(fact_6073_min_Oleft__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) ) ).

% min.left_idem
tff(fact_6074_min_Oidem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),A3) = A3 ) ).

% min.idem
tff(fact_6075_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C3))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3) ) ) ) ).

% min.bounded_iff
tff(fact_6076_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = B3 ) ) ) ).

% min.absorb2
tff(fact_6077_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = A3 ) ) ) ).

% min.absorb1
tff(fact_6078_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = A3 ) ) ) ).

% min.absorb3
tff(fact_6079_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = B3 ) ) ) ).

% min.absorb4
tff(fact_6080_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Xa)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Z),Ya) ) ) ) ).

% min_less_iff_conj
tff(fact_6081_min__top2,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),top_top(A)) = Xa ) ).

% min_top2
tff(fact_6082_min__top,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),top_top(A)),Xa) = Xa ) ).

% min_top
tff(fact_6083_min__bot,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),bot_bot(A)),Xa) = bot_bot(A) ) ).

% min_bot
tff(fact_6084_min__bot2,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),bot_bot(A)) = bot_bot(A) ) ).

% min_bot2
tff(fact_6085_max__min__same_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ya: A,Xa: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Ya),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)) = Ya ) ).

% max_min_same(4)
tff(fact_6086_max__min__same_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)),Ya) = Ya ) ).

% max_min_same(3)
tff(fact_6087_max__min__same_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)),Xa) = Xa ) ).

% max_min_same(2)
tff(fact_6088_max__min__same_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)) = Xa ) ).

% max_min_same(1)
tff(fact_6089_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V2)) ) ).

% min_number_of(1)
tff(fact_6090_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Xa)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_6091_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xa)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_6092_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_6093_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_6094_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),Xa)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_6095_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),Xa)),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_6096_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ).

% min_number_of(2)
tff(fact_6097_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V2)) ) ).

% min_number_of(3)
tff(fact_6098_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V2: num] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V2))) ) ).

% min_number_of(4)
tff(fact_6099_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ).

% Min_insert
tff(fact_6100_Min_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) = aa(set(A),A,lattic643756798350308766er_Min(A),A4) ) ) ) ) ).

% Min.in_idem
tff(fact_6101_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,B3: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),D2)) ) ) ) ).

% min.mono
tff(fact_6102_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) ) ) ) ).

% min.orderE
tff(fact_6103_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3) ) ) ).

% min.orderI
tff(fact_6104_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3) ) ) ) ).

% min.boundedE
tff(fact_6105_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C3)) ) ) ) ).

% min.boundedI
tff(fact_6106_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) ) ) ) ).

% min.order_iff
tff(fact_6107_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),A3) ) ).

% min.cobounded1
tff(fact_6108_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),B3) ) ).

% min.cobounded2
tff(fact_6109_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = A3 ) ) ) ).

% min.absorb_iff1
tff(fact_6110_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = B3 ) ) ) ).

% min.absorb_iff2
tff(fact_6111_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),C3) ) ) ).

% min.coboundedI1
tff(fact_6112_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),C3) ) ) ).

% min.coboundedI2
tff(fact_6113_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_6114_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Ya: A,Xa: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ya),Xa)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya) = Ya ) ) ) ).

% min_absorb2
tff(fact_6115_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Ya: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya) = Xa ) ) ) ).

% min_absorb1
tff(fact_6116_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A3),B3),A3,B3) ) ).

% min_def
tff(fact_6117_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa2: A] :
          aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Xa2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa2),X2,Xa2) ) ).

% min_def_raw
tff(fact_6118_max__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_max(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)) ) ) ) ).

% max_of_antimono
tff(fact_6119_min__of__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),Xa: A,Ya: A] :
          ( order_antimono(A,B,F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya)) ) ) ) ).

% min_of_antimono
tff(fact_6120_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Ya: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Ya)) ) ).

% minus_min_eq_max
tff(fact_6121_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [Xa: A,Ya: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),Xa)),aa(A,A,uminus_uminus(A),Ya)) ) ).

% minus_max_eq_min
tff(fact_6122_inf__min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( inf_inf(A) = ord_min(A) ) ) ).

% inf_min
tff(fact_6123_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Ya: 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),Xa),Ya)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Ya),Z)) ) ).

% min_add_distrib_left
tff(fact_6124_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),aa(A,A,aa(A,fun(A,A),ord_min(A),Ya),Z)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Ya)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),Z)) ) ).

% min_add_distrib_right
tff(fact_6125_nat__mult__min__left,axiom,
    ! [M: nat,Na: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)),Q5) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Na),Q5)) ).

% nat_mult_min_left
tff(fact_6126_nat__mult__min__right,axiom,
    ! [M: nat,Na: nat,Q5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Na),Q5)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Na)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q5)) ).

% nat_mult_min_right
tff(fact_6127_linorder_OMax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : lattices_Max(A,Less_eq) = lattices_Max(A,Less_eq) ).

% linorder.Max.cong
tff(fact_6128_min_Oleft__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),B3),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C3)) ) ).

% min.left_commute
tff(fact_6129_min_Ocommute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) = aa(A,A,aa(A,fun(A,A),ord_min(A),B3),A3) ) ).

% min.commute
tff(fact_6130_min_Oassoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),C3) = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C3)) ) ).

% min.assoc
tff(fact_6131_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xa: nat,Ya: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Xa),Ya)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),Xa)),aa(nat,A,semiring_1_of_nat(A),Ya)) ) ).

% of_nat_min
tff(fact_6132_min__max__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),A3),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),C3)) ) ).

% min_max_distrib2
tff(fact_6133_min__max__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C3: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),C3)),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),A3)),aa(A,A,aa(A,fun(A,A),ord_min(A),C3),A3)) ) ).

% min_max_distrib1
tff(fact_6134_max__min__distrib2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),B3)),aa(A,A,aa(A,fun(A,A),ord_max(A),A3),C3)) ) ).

% max_min_distrib2
tff(fact_6135_max__min__distrib1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C3: A,A3: A] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C3)),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B3),A3)),aa(A,A,aa(A,fun(A,A),ord_max(A),C3),A3)) ) ).

% max_min_distrib1
tff(fact_6136_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xa: A,Ya: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,minus_minus(A,Xa),Z)),aa(A,A,minus_minus(A,Ya),Z)) ) ).

% min_diff_distrib_left
tff(fact_6137_min__diff,axiom,
    ! [M: nat,I: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,minus_minus(nat,M),I)),aa(nat,nat,minus_minus(nat,Na),I)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),Na)),I) ).

% min_diff
tff(fact_6138_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_6139_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(A,A,aa(A,fun(A,A),ord_min(A),B3),C3))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3) ) ) ) ).

% min.strict_boundedE
tff(fact_6140_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),B3)
        <=> ( ( A3 = aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
tff(fact_6141_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),C3) ) ) ).

% min.strict_coboundedI1
tff(fact_6142_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C3: A,A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),B3),C3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A3),B3)),C3) ) ) ).

% min.strict_coboundedI2
tff(fact_6143_min__of__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F2: fun(A,B),M: A,Na: A] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( aa(B,B,aa(B,fun(B,B),ord_min(B),aa(A,B,F2,M)),aa(A,B,F2,Na)) = aa(A,B,F2,aa(A,A,aa(A,fun(A,A),ord_min(A),M),Na)) ) ) ) ).

% min_of_mono
tff(fact_6144_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),P3))) ) ).

% min_mult_distrib_right
tff(fact_6145_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xa: A,Ya: A,P3: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya)),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xa),P3)),aa(A,A,aa(A,fun(A,A),times_times(A),Ya),P3))) ) ).

% max_mult_distrib_right
tff(fact_6146_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xa: A,Ya: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Ya)),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Ya))) ) ).

% min_mult_distrib_left
tff(fact_6147_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P3: A,Xa: A,Ya: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Ya)),aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Xa)),aa(A,A,aa(A,fun(A,A),times_times(A),P3),Ya))) ) ).

% max_mult_distrib_left
tff(fact_6148_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A,P3: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),Xa),Ya),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xa,P3)),divide_divide(A,Ya,P3)),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xa,P3)),divide_divide(A,Ya,P3))) ) ).

% max_divide_distrib_right
tff(fact_6149_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xa: A,Ya: A,P3: A] :
          divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),Ya),P3) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),P3),aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,Xa,P3)),divide_divide(A,Ya,P3)),aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,Xa,P3)),divide_divide(A,Ya,P3))) ) ).

% min_divide_distrib_right
tff(fact_6150_Inf__insert__finite,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),S)) = $ite(S = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,complete_Inf_Inf(A),S))) ) ) ) ).

% Inf_insert_finite
tff(fact_6151_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N4: set(A)] :
          ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X3)),aa(A,A,H,Y))
         => ( aa(set(A),$o,finite_finite2(A),N4)
           => ( ( N4 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798350308766er_Min(A),N4)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_6152_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( B2 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B2)),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) = aa(set(A),A,lattic643756798350308766er_Min(A),A4) ) ) ) ) ) ).

% Min.subset
tff(fact_6153_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))
             => member(A,aa(set(A),A,lattic643756798350308766er_Min(A),A4),A4) ) ) ) ) ).

% Min.closed
tff(fact_6154_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( ( A4 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),A4)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_6155_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( ( B2 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),aa(set(A),A,lattic643756798350308766er_Min(A),B2)) ) ) ) ) ) ) ).

% Min.union
tff(fact_6156_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = finite_fold(A,A,ord_min(A),Xa,A4) ) ) ) ).

% Min.eq_fold
tff(fact_6157_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% Min.insert_remove
tff(fact_6158_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xa: A] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A4) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),ord_min(A),Xa),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% Min.remove
tff(fact_6159_lexord__take__index__conv,axiom,
    ! [A: $tType,Xa: list(A),Ya: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xa),Ya),lexord(A,R2))
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(list(A),nat,size_size(list(A)),Ya))
          & ( take(A,aa(list(A),nat,size_size(list(A)),Xa),Ya) = Xa ) )
        | ? [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xa)),aa(list(A),nat,size_size(list(A)),Ya)))
            & ( take(A,I4,Xa) = take(A,I4,Ya) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xa),I4)),aa(nat,A,nth(A,Ya),I4)),R2) ) ) ) ).

% lexord_take_index_conv
tff(fact_6160_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_ta(A,fun(A,$o))) = ord_min(A) ) ) ).

% dual_max
tff(fact_6161_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_6162_ord_Omax__def,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),A3: A,B3: A] :
      aa(A,A,aa(A,fun(A,A),max(A,Less_eq),A3),B3) = $ite(aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),B3),B3,A3) ).

% ord.max_def
tff(fact_6163_ord_Omax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o))] : max(A,Less_eq) = max(A,Less_eq) ).

% ord.max.cong
tff(fact_6164_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_6165_find__Some__iff2,axiom,
    ! [A: $tType,Xa: A,P: fun(A,$o),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),Xa) = find(A,P,Xs) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),I4))
          & ( Xa = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff2
tff(fact_6166_hd__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( Na != zero_zero(nat) )
     => ( hd(A,replicate(A,Na,Xa)) = Xa ) ) ).

% hd_replicate
tff(fact_6167_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),J)
     => ( hd(A,take(A,J,Xs)) = hd(A,Xs) ) ) ).

% hd_take
tff(fact_6168_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_6169_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Xa: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),Xa) )
    <=> ? [I4: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(A,$o,P,aa(nat,A,nth(A,Xs),I4))
          & ( Xa = aa(nat,A,nth(A,Xs),I4) )
          & ! [J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),I4)
             => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% find_Some_iff
tff(fact_6170_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F2: fun(A,nat),Xs: list(A)] :
      size_list(A,F2,Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F2,hd(A,Xs))),size_list(A,F2,tl(A,Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_6171_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_6172_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),tl(A,Xs)))) ).

% Nitpick.size_list_simp(2)
tff(fact_6173_nth__tl,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),tl(A,Xs)))
     => ( aa(nat,A,nth(A,tl(A,Xs)),Na) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,Na)) ) ) ).

% nth_tl
tff(fact_6174_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( sgn_sgn(complex,Z) = cis(arg(Z)) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_6175_nth__rotate,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,M),Xs)),Na) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_6176_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => ( aa(list(A),list(A),rotate(A,Na),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_6177_rotate__id,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,Na,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),rotate(A,Na),Xs) = Xs ) ) ).

% rotate_id
tff(fact_6178_cis__Arg__unique,axiom,
    ! [Z: complex,Xa: real] :
      ( ( sgn_sgn(complex,Z) = cis(Xa) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Xa)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),pi)
         => ( arg(Z) = Xa ) ) ) ) ).

% cis_Arg_unique
tff(fact_6179_bij__betw__roots__unity,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => bij_betw(nat,complex,aTP_Lamp_abq(nat,fun(nat,complex),Na),aa(nat,set(nat),set_ord_lessThan(nat),Na),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_au(nat,fun(complex,$o),Na))) ) ).

% bij_betw_roots_unity
tff(fact_6180_bij__betw__nth__root__unity,axiom,
    ! [C3: complex,Na: nat] :
      ( ( C3 != zero_zero(complex) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
       => 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(Na),real_V7770717601297561774m_norm(complex,C3)))),cis(divide_divide(real,arg(C3),aa(nat,real,semiring_1_of_nat(real),Na))))),aa(fun(complex,$o),set(complex),collect(complex),aTP_Lamp_au(nat,fun(complex,$o),Na)),aa(fun(complex,$o),set(complex),collect(complex),aa(nat,fun(complex,$o),aTP_Lamp_av(complex,fun(nat,fun(complex,$o)),C3),Na))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_6181_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),A14: set(B),B2: set(A),B13: set(B)] :
      ( bij_betw(A,B,F2,A4,A14)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
       => ( ( aa(set(A),set(B),image2(A,B,F2),B2) = B13 )
         => bij_betw(A,B,F2,B2,B13) ) ) ) ).

% bij_betw_subset
tff(fact_6182_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F9: fun(B,A),F2: fun(A,B),A14: set(B)] :
      ( ! [X3: A] :
          ( member(A,X3,A4)
         => ( aa(B,A,F9,aa(A,B,F2,X3)) = X3 ) )
     => ( ! [X3: B] :
            ( member(B,X3,A14)
           => ( aa(A,B,F2,aa(B,A,F9,X3)) = X3 ) )
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),A14)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F9),A14)),A4)
           => bij_betw(A,B,F2,A4,A14) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_6183_bij__betw__empty1,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B)] :
      ( bij_betw(A,B,F2,bot_bot(set(A)),A4)
     => ( A4 = bot_bot(set(B)) ) ) ).

% bij_betw_empty1
tff(fact_6184_bij__betw__empty2,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( bij_betw(A,B,F2,A4,bot_bot(set(B)))
     => ( A4 = bot_bot(set(A)) ) ) ).

% bij_betw_empty2
tff(fact_6185_bij__betw__same__card,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B2: set(B)] :
      ( bij_betw(A,B,F2,A4,B2)
     => ( aa(set(A),nat,finite_card(A),A4) = aa(set(B),nat,finite_card(B),B2) ) ) ).

% bij_betw_same_card
tff(fact_6186_bij__betw__funpow,axiom,
    ! [A: $tType,F2: fun(A,A),S: set(A),Na: 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)),Na),F2),S,S) ) ).

% bij_betw_funpow
tff(fact_6187_bij__betw__iff__card,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( ? [F7: fun(A,B)] : bij_betw(A,B,F7,A4,B2)
        <=> ( aa(set(A),nat,finite_card(A),A4) = aa(set(B),nat,finite_card(B),B2) ) ) ) ) ).

% bij_betw_iff_card
tff(fact_6188_finite__same__card__bij,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( ( aa(set(A),nat,finite_card(A),A4) = aa(set(B),nat,finite_card(B),B2) )
         => ? [H4: fun(A,B)] : bij_betw(A,B,H4,A4,B2) ) ) ) ).

% finite_same_card_bij
tff(fact_6189_bij__fn,axiom,
    ! [A: $tType,F2: fun(A,A),Na: 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)),Na),F2),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_6190_bij__betw__finite,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B2: set(B)] :
      ( bij_betw(A,B,F2,A4,B2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
      <=> aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% bij_betw_finite
tff(fact_6191_bij__betwI_H,axiom,
    ! [A: $tType,B: $tType,X4: set(A),F2: fun(A,B),Y3: set(B)] :
      ( ! [X3: A] :
          ( member(A,X3,X4)
         => ! [Y: A] :
              ( member(A,Y,X4)
             => ( ( aa(A,B,F2,X3) = aa(A,B,F2,Y) )
              <=> ( X3 = Y ) ) ) )
     => ( ! [X3: A] :
            ( member(A,X3,X4)
           => member(B,aa(A,B,F2,X3),Y3) )
       => ( ! [Y: B] :
              ( member(B,Y,Y3)
             => ? [X2: A] :
                  ( member(A,X2,X4)
                  & ( Y = aa(A,B,F2,X2) ) ) )
         => bij_betw(A,B,F2,X4,Y3) ) ) ) ).

% bij_betwI'
tff(fact_6192_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F9: fun(A,B),A14: set(A),A16: set(B),F2: fun(C,A),A4: set(C)] :
      ( bij_betw(A,B,F9,A14,A16)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image2(C,A,F2),A4)),A14)
       => ( bij_betw(C,A,F2,A4,A14)
        <=> bij_betw(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,F9),F2),A4,A16) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_6193_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B2: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),B2)
       => ( inj_on(B,A,G,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G),B2)),A4)
           => ? [H4: fun(A,B)] : bij_betw(A,B,H4,A4,B2) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_6194_notIn__Un__bij__betw,axiom,
    ! [A: $tType,B: $tType,B3: A,A4: set(A),F2: fun(A,B),A14: set(B)] :
      ( ~ member(A,B3,A4)
     => ( ~ member(B,aa(A,B,F2,B3),A14)
       => ( bij_betw(A,B,F2,A4,A14)
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A14),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,B3)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw
tff(fact_6195_notIn__Un__bij__betw3,axiom,
    ! [A: $tType,B: $tType,B3: A,A4: set(A),F2: fun(A,B),A14: set(B)] :
      ( ~ member(A,B3,A4)
     => ( ~ member(B,aa(A,B,F2,B3),A14)
       => ( bij_betw(A,B,F2,A4,A14)
        <=> bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A14),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,F2,B3)),bot_bot(set(B))))) ) ) ) ).

% notIn_Un_bij_betw3
tff(fact_6196_bij__betw__combine,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),B2: set(B),C2: set(A),D3: set(B)] :
      ( bij_betw(A,B,F2,A4,B2)
     => ( bij_betw(A,B,F2,C2,D3)
       => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B2),D3) = bot_bot(set(B)) )
         => bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B2),D3)) ) ) ) ).

% bij_betw_combine
tff(fact_6197_bij__betw__partition,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),C2: set(A),B2: set(B),D3: set(B)] :
      ( bij_betw(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),B2),D3))
     => ( bij_betw(A,B,F2,C2,D3)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),C2) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),B2),D3) = bot_bot(set(B)) )
           => bij_betw(A,B,F2,A4,B2) ) ) ) ) ).

% bij_betw_partition
tff(fact_6198_bij__betw__disjoint__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(A),C2: set(B),G: fun(A,B),B2: set(A),D3: set(B)] :
      ( bij_betw(A,B,F2,A4,C2)
     => ( bij_betw(A,B,G,B2,D3)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) )
         => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),C2),D3) = bot_bot(set(B)) )
           => bij_betw(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_yq(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),F2),A4),G),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),C2),D3)) ) ) ) ) ).

% bij_betw_disjoint_Un
tff(fact_6199_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A4: fun(A,set(B)),F2: fun(B,C),A14: fun(A,set(C))] :
      ( ! [I2: A,J2: A] :
          ( member(A,I2,I5)
         => ( member(A,J2,I5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A4,I2)),aa(A,set(B),A4,J2))
              | aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),A4,J2)),aa(A,set(B),A4,I2)) ) ) )
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => bij_betw(B,C,F2,aa(A,set(B),A4,I2),aa(A,set(C),A14,I2)) )
       => bij_betw(B,C,F2,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),A14),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_6200_infinite__imp__bij__betw2,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ? [H4: fun(A,A)] : bij_betw(A,A,H4,A4,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw2
tff(fact_6201_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ? [H4: fun(A,A)] : bij_betw(A,A,H4,A4,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_6202_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M5)
     => ? [H4: fun(nat,A)] : bij_betw(nat,A,H4,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M5)),M5) ) ).

% ex_bij_betw_nat_finite
tff(fact_6203_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M5)
     => ? [H4: fun(nat,A)] : bij_betw(nat,A,H4,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M5)),M5) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_6204_finite__bij__enumerate,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => bij_betw(nat,A,infini527867602293511546merate(A,S),aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),S)),S) ) ) ).

% finite_bij_enumerate
tff(fact_6205_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S6: set(A),T6: set(B),H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S6)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S),S6),aa(set(B),set(B),minus_minus(set(B),T3),T6))
             => ( ! [A5: A] :
                    ( member(A,A5,S6)
                   => ( aa(B,C,G,aa(A,B,H,A5)) = zero_zero(C) ) )
               => ( ! [B5: B] :
                      ( member(B,B5,T6)
                     => ( aa(B,C,G,B5) = zero_zero(C) ) )
                 => ( 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_abr(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),G),T3) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_6206_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S6: set(A),T6: set(B),H: fun(A,B),S: set(A),T3: set(B),G: fun(B,C)] :
          ( aa(set(A),$o,finite_finite2(A),S6)
         => ( aa(set(B),$o,finite_finite2(B),T6)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S),S6),aa(set(B),set(B),minus_minus(set(B),T3),T6))
             => ( ! [A5: A] :
                    ( member(A,A5,S6)
                   => ( aa(B,C,G,aa(A,B,H,A5)) = one_one(C) ) )
               => ( ! [B5: B] :
                      ( member(B,B5,T6)
                     => ( aa(B,C,G,B5) = one_one(C) ) )
                 => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(B,C),fun(A,C),aTP_Lamp_abs(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),G),T3) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_6207_ex__bij__betw__strict__mono__card,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [M5: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),M5)
         => ~ ! [H4: fun(nat,A)] :
                ( bij_betw(nat,A,H4,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),M5)),M5)
               => ~ strict_mono_on(nat,A,H4,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),M5))) ) ) ) ).

% ex_bij_betw_strict_mono_card
tff(fact_6208_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_abt(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_6209_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( linord4507533701916653071of_set(A,A4) = aa(list(A),list(A),cons(A,aa(set(A),A,lattic643756798350308766er_Min(A),A4)),linord4507533701916653071of_set(A,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(set(A),A,lattic643756798350308766er_Min(A),A4)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_6210_bij__betw__Suc,axiom,
    ! [M5: set(nat),N4: set(nat)] :
      ( bij_betw(nat,nat,suc,M5,N4)
    <=> ( aa(set(nat),set(nat),image2(nat,nat,suc),M5) = N4 ) ) ).

% bij_betw_Suc
tff(fact_6211_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,X21),X22)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X21),aa(list(A),set(A),set2(A),X22)) ).

% list.simps(15)
tff(fact_6212_nth__Cons__0,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),zero_zero(nat)) = Xa ).

% nth_Cons_0
tff(fact_6213_n__lists__Nil,axiom,
    ! [A: $tType,Na: nat] :
      n_lists(A,Na,nil(A)) = $ite(Na = zero_zero(nat),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))),nil(list(A))) ).

% n_lists_Nil
tff(fact_6214_nths__singleton,axiom,
    ! [A: $tType,Xa: A,A4: set(nat)] :
      nths(A,aa(list(A),list(A),cons(A,Xa),nil(A)),A4) = $ite(member(nat,zero_zero(nat),A4),aa(list(A),list(A),cons(A,Xa),nil(A)),nil(A)) ).

% nths_singleton
tff(fact_6215_nth__Cons__pos,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Na) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6216_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys2))
     => ( Xs != aa(list(A),list(A),cons(A,Xa),Ys2) ) ) ).

% impossible_Cons
tff(fact_6217_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),Xa: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xa),Xs))) ).

% set_subset_Cons
tff(fact_6218_insort__key_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Ya: A,Ys2: list(A)] :
          linorder_insort_key(A,B,F2,Xa,aa(list(A),list(A),cons(A,Ya),Ys2)) = $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Ya)),aa(list(A),list(A),cons(A,Xa),aa(list(A),list(A),cons(A,Ya),Ys2)),aa(list(A),list(A),cons(A,Ya),linorder_insort_key(A,B,F2,Xa,Ys2))) ) ).

% insort_key.simps(2)
tff(fact_6219_list__update__code_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Ya: A] : list_update(A,aa(list(A),list(A),cons(A,Xa),Xs),zero_zero(nat),Ya) = aa(list(A),list(A),cons(A,Ya),Xs) ).

% list_update_code(2)
tff(fact_6220_some__in__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( member(A,fChoice(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)),A4)
    <=> ( A4 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_6221_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Ya: A,Xs: list(A),Ys2: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),cons(A,Ya)),shuffles(A,Xs,Ys2))),shuffles(A,Xs,aa(list(A),list(A),cons(A,Ya),Ys2))) ).

% Cons_shuffles_subset2
tff(fact_6222_Cons__shuffles__subset1,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Ys2: list(A)] : aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image2(list(A),list(A),cons(A,Xa)),shuffles(A,Xs,Ys2))),shuffles(A,aa(list(A),list(A),cons(A,Xa),Xs),Ys2)) ).

% Cons_shuffles_subset1
tff(fact_6223_Suc__le__length__iff,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,X),Ys4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Ys4)) ) ) ).

% Suc_le_length_iff
tff(fact_6224_insort__is__Cons,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F2: fun(A,B),A3: A] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),aa(A,B,F2,X3)) )
         => ( linorder_insort_key(A,B,F2,A3,Xs) = aa(list(A),list(A),cons(A,A3),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_6225_bij__enumerate,axiom,
    ! [S: set(nat)] :
      ( ~ aa(set(nat),$o,finite_finite2(nat),S)
     => bij_betw(nat,nat,infini527867602293511546merate(nat,S),top_top(set(nat)),S) ) ).

% bij_enumerate
tff(fact_6226_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_6227_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M5: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),M5)
     => ? [H4: fun(A,nat)] : bij_betw(A,nat,H4,M5,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M5))) ) ).

% ex_bij_betw_finite_nat
tff(fact_6228_the__elem__set,axiom,
    ! [A: $tType,Xa: A] : the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xa),nil(A)))) = Xa ).

% the_elem_set
tff(fact_6229_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => aa(fun(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),$o,order_mono(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),aTP_Lamp_abu(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ).

% lexordp.mono
tff(fact_6230_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_6231_nth__Cons_H,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Na) = $ite(Na = zero_zero(nat),Xa,aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ).

% nth_Cons'
tff(fact_6232_remdups__adj__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      remdups_adj(A,replicate(A,Na,Xa)) = $ite(Na = zero_zero(nat),nil(A),aa(list(A),list(A),cons(A,Xa),nil(A))) ).

% remdups_adj_replicate
tff(fact_6233_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xa: fun(A,nat),X21: A,X22: list(A)] : size_list(A,Xa,aa(list(A),list(A),cons(A,X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xa,X21)),size_list(A,Xa,X22))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6234_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or3652927894154168847AtMost(nat,I,J)) = aa(list(nat),list(nat),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_6235_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I)),J)
     => ( linord4507533701916653071of_set(nat,set_or5935395276787703475ssThan(nat,I,J)) = aa(list(nat),list(nat),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_6236_nth__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat] :
      ( ~ member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Na) = Xa )
        <=> ( Na = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6237_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xa: A,Ya: A,Xs: list(A),Na: nat] :
      ( ( Xa != Ya )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xa),Xs)),Na) = Ya )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Na),one_one(nat))) = Ya )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6238_take__Cons_H,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] :
      take(A,Na,aa(list(A),list(A),cons(A,Xa),Xs)) = $ite(Na = zero_zero(nat),nil(A),aa(list(A),list(A),cons(A,Xa),take(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),Xs))) ).

% take_Cons'
tff(fact_6239_Cons__replicate__eq,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),Na: nat,Ya: A] :
      ( ( aa(list(A),list(A),cons(A,Xa),Xs) = replicate(A,Na,Ya) )
    <=> ( ( Xa = Ya )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
        & ( Xs = replicate(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),Xa) ) ) ) ).

% Cons_replicate_eq
tff(fact_6240_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),Na: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),cons(A,M),Ms)),aa(list(A),list(A),cons(A,Na),Ns)),lenlex(A,R2))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,M),Na),R2) )
        | ( ( M = Na )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ms),Ns),lenlex(A,R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6241_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( lattic7623131987881927897min_on(A,B,F2,S) = fChoice(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_abv(set(A),fun(fun(A,B),fun(A,$o)),S),F2)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_6242_Pow__set_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] :
      pow2(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xa),Xs))) = $let(
        a2: set(set(A)),
        a2:= pow2(A,aa(list(A),set(A),set2(A),Xs)),
        aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),a2),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa)),a2)) ) ).

% Pow_set(2)
tff(fact_6243_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xa: A,Ys2: list(A)] : aa(nat,A,nth(A,append(A,Xs,append(A,aa(list(A),list(A),cons(A,Xa),nil(A)),Ys2))),aa(list(A),nat,size_size(list(A)),Xs)) = Xa ).

% concat_inth
tff(fact_6244_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I),Js,upto_aux(I,aa(int,int,minus_minus(int,J),one_one(int)),aa(list(int),list(int),cons(int,J),Js))) ).

% upto_aux_rec
tff(fact_6245_set__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] : aa(list(A),set(A),set2(A),append(A,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) ).

% set_append
tff(fact_6246_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( distinct(A,append(A,Xs,Ys2))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys2)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_6247_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys2: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,append(A,Xs,Ys2),I,Xa) = append(A,list_update(A,Xs,I,Xa),Ys2) ) ) ).

% list_update_append1
tff(fact_6248_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Na: nat] :
      aa(nat,A,nth(A,append(A,Xs,Ys2)),Na) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,A,nth(A,Xs),Na),aa(nat,A,nth(A,Ys2),aa(nat,nat,minus_minus(nat,Na),aa(list(A),nat,size_size(list(A)),Xs)))) ).

% nth_append
tff(fact_6249_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Na: nat,Xa: A] :
      list_update(A,append(A,Xs,Ys2),Na,Xa) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs)),append(A,list_update(A,Xs,Na,Xa),Ys2),append(A,Xs,list_update(A,Ys2,aa(nat,nat,minus_minus(nat,Na),aa(list(A),nat,size_size(list(A)),Xs)),Xa))) ).

% list_update_append
tff(fact_6250_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys2 != nil(A) )
       => ( ( append(A,Xs,Ys2) = append(A,Ys2,Xs) )
         => ? [N2: nat,Zs2: list(A)] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),one_one(nat)),N2)
              & ( concat(A,replicate(list(A),N2,Zs2)) = append(A,Xs,Ys2) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_6251_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W2: list(A),R2: set(product_prod(A,A)),V2: list(A),Z: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),U),W2),lexord(A,R2))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W2)),aa(list(A),nat,size_size(list(A)),U))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),append(A,U,V2)),append(A,W2,Z)),lexord(A,R2)) ) ) ).

% lexord_sufI
tff(fact_6252_nths__Cons,axiom,
    ! [A: $tType,Xa: A,L: list(A),A4: set(nat)] :
      nths(A,aa(list(A),list(A),cons(A,Xa),L),A4) = append(A,
        $ite(member(nat,zero_zero(nat),A4),aa(list(A),list(A),cons(A,Xa),nil(A)),nil(A)),
        nths(A,L,aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_abw(set(nat),fun(nat,$o),A4)))) ).

% nths_Cons
tff(fact_6253_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,aa(nat,nat,suc,I),Xs) = append(A,take(A,I,Xs),aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_6254_nth__repl,axiom,
    ! [A: $tType,M: nat,Xs: list(A),Na: nat,Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( M != Na )
         => ( aa(nat,A,nth(A,append(A,take(A,Na,Xs),append(A,aa(list(A),list(A),cons(A,Xa),nil(A)),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),one_one(nat)),Xs)))),M) = aa(nat,A,nth(A,Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_6255_pos__n__replace,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),Ya: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),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,Na,Xs),append(A,aa(list(A),list(A),cons(A,Ya),nil(A)),drop(A,aa(nat,nat,suc,Na),Xs)))) ) ) ).

% pos_n_replace
tff(fact_6256_drop0,axiom,
    ! [A: $tType,X2: list(A)] : drop(A,zero_zero(nat),X2) = X2 ).

% drop0
tff(fact_6257_drop__update__cancel,axiom,
    ! [A: $tType,Na: nat,M: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
     => ( drop(A,M,list_update(A,Xs,Na,Xa)) = drop(A,M,Xs) ) ) ).

% drop_update_cancel
tff(fact_6258_drop__eq__Nil2,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,Na,Xs) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na) ) ).

% drop_eq_Nil2
tff(fact_6259_drop__eq__Nil,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( ( drop(A,Na,Xs) = nil(A) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na) ) ).

% drop_eq_Nil
tff(fact_6260_drop__all,axiom,
    ! [A: $tType,Xs: list(A),Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),Na)
     => ( drop(A,Na,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_6261_nth__drop,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,drop(A,Na,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Na),I)) ) ) ).

% nth_drop
tff(fact_6262_set__drop__subset,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,Na,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% set_drop_subset
tff(fact_6263_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_6264_drop__eq__nths,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : drop(A,Na,Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),Na))) ).

% drop_eq_nths
tff(fact_6265_set__drop__subset__set__drop,axiom,
    ! [A: $tType,Na: nat,M: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,M,Xs))),aa(list(A),set(A),set2(A),drop(A,Na,Xs))) ) ).

% set_drop_subset_set_drop
tff(fact_6266_drop__update__swap,axiom,
    ! [A: $tType,M: nat,Na: nat,Xs: list(A),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( drop(A,M,list_update(A,Xs,Na,Xa)) = list_update(A,drop(A,M,Xs),aa(nat,nat,minus_minus(nat,Na),M),Xa) ) ) ).

% drop_update_swap
tff(fact_6267_drop__Cons_H,axiom,
    ! [A: $tType,Na: nat,Xa: A,Xs: list(A)] :
      drop(A,Na,aa(list(A),list(A),cons(A,Xa),Xs)) = $ite(Na = zero_zero(nat),aa(list(A),list(A),cons(A,Xa),Xs),drop(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),Xs)) ).

% drop_Cons'
tff(fact_6268_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( append(A,Xs_1,Xs_2) = append(A,Ys_1,Ys_2) )
    <=> $ite(
          aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)),
          ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
          & ( Xs_2 = append(A,drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1),Ys_2) ) ),
          ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
          & ( append(A,drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1),Xs_2) = Ys_2 ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_6269_hd__drop__conv__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( hd(A,drop(A,Na,Xs)) = aa(nat,A,nth(A,Xs),Na) ) ) ).

% hd_drop_conv_nth
tff(fact_6270_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),drop(A,aa(nat,nat,suc,I),Xs)) = drop(A,I,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_6271_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I: nat,J: nat] :
      ( distinct(A,Vs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),take(A,I,Vs))),aa(list(A),set(A),set2(A),drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_6272_id__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( Xs = append(A,take(A,I,Xs),aa(list(A),list(A),cons(A,aa(nat,A,nth(A,Xs),I)),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_6273_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I: nat,Xs: list(A),A3: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,Xs,I,A3) = append(A,take(A,I,Xs),aa(list(A),list(A),cons(A,A3),drop(A,aa(nat,nat,suc,I),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_6274_take__hd__drop,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( append(A,take(A,Na,Xs),aa(list(A),list(A),cons(A,hd(A,drop(A,Na,Xs))),nil(A))) = take(A,aa(nat,nat,suc,Na),Xs) ) ) ).

% take_hd_drop
tff(fact_6275_upto_Opelims,axiom,
    ! [Xa: int,Xaa: int,Ya: list(int)] :
      ( ( upto(Xa,Xaa) = Ya )
     => ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,Xa),Xaa))
       => ~ ( ( Ya = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa),aa(list(int),list(int),cons(int,Xa),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int)),Xaa)),nil(int)) )
           => ~ aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,Xa),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_6276_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_6277_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil
tff(fact_6278_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I) ) ).

% upto_Nil2
tff(fact_6279_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),J),I)
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_6280_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),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_6281_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)),aa(list(int),list(int),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),Na))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_6282_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Na: num] :
      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),Na))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))),aa(list(int),list(int),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),Na)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_6283_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Na)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),Na)),aa(list(int),list(int),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),Na))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_6284_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Na: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Na))),aa(list(int),list(int),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),Na)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_6285_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),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_6286_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_6287_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_6288_upto_Oelims,axiom,
    ! [Xa: int,Xaa: int,Ya: list(int)] :
      ( ( upto(Xa,Xaa) = Ya )
     => ( Ya = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa),aa(list(int),list(int),cons(int,Xa),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_6289_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6290_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),aa(list(int),list(int),cons(int,J),nil(int))) ) ) ).

% upto_rec2
tff(fact_6291_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),aa(list(int),list(int),cons(int,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_6292_to__nat__on__def,axiom,
    ! [A: $tType,S: set(A)] : countable_to_nat_on(A,S) = fChoice(fun(A,nat),aTP_Lamp_abx(set(A),fun(fun(A,nat),$o),S)) ).

% to_nat_on_def
tff(fact_6293_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),M: fun(A,option(B)),Xa: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys2))
     => ( map_upds(A,B,M,append(A,Xs,aa(list(A),list(A),cons(A,Xa),nil(A))),Ys2) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys2),Xa,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_6294_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I: nat,M: fun(A,option(B)),Ys2: list(B),Ya: B] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( map_upds(A,B,M,Xs,list_update(B,Ys2,I,Ya)) = map_upds(A,B,M,Xs,Ys2) ) ) ).

% map_upds_list_update2_drop
tff(fact_6295_to__nat__on__finite,axiom,
    ! [A: $tType,S: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => bij_betw(A,nat,countable_to_nat_on(A,S),S,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),S))) ) ).

% to_nat_on_finite
tff(fact_6296_restrict__upd__same,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xa: A,Ya: B] : restrict_map(A,B,fun_upd(A,option(B),M,Xa,aa(B,option(B),some(B),Ya)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))) = restrict_map(A,B,M,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))) ).

% restrict_upd_same
tff(fact_6297_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),D3: set(A),M: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D3)
       => ( restrict_map(A,B,map_upds(A,B,M,Xs,Ys2),D3) = map_upds(A,B,restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D3),aa(list(A),set(A),set2(A),Xs))),Xs,Ys2) ) ) ) ).

% restrict_map_upds
tff(fact_6298_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),X2: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X2) = none(B) ).

% restrict_map_to_empty
tff(fact_6299_restrict__fun__upd,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xa: A,Ya: option(B),D3: set(A)] :
      restrict_map(A,B,fun_upd(A,option(B),M,Xa,Ya),D3) = $ite(member(A,Xa,D3),fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),Xa,Ya),restrict_map(A,B,M,D3)) ).

% restrict_fun_upd
tff(fact_6300_fun__upd__restrict__conv,axiom,
    ! [A: $tType,B: $tType,Xa: A,D3: set(A),M: fun(A,option(B)),Ya: option(B)] :
      ( member(A,Xa,D3)
     => ( fun_upd(A,option(B),restrict_map(A,B,M,D3),Xa,Ya) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),Xa,Ya) ) ) ).

% fun_upd_restrict_conv
tff(fact_6301_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),D3: set(A),Xa: A] :
      fun_upd(A,option(B),restrict_map(A,B,M,D3),Xa,none(B)) = $ite(member(A,Xa,D3),restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),restrict_map(A,B,M,D3)) ).

% fun_upd_None_restrict
tff(fact_6302_restrict__map__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),A3: A,A4: set(A)] : restrict_map(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = fun_upd(A,option(B),restrict_map(A,B,F2,A4),A3,aa(A,option(B),F2,A3)) ).

% restrict_map_insert
tff(fact_6303_fun__upd__restrict,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),D3: set(A),Xa: A,Ya: option(B)] : fun_upd(A,option(B),restrict_map(A,B,M,D3),Xa,Ya) = fun_upd(A,option(B),restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),Xa,Ya) ).

% fun_upd_restrict
tff(fact_6304_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xa: A] : restrict_map(A,B,F2,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))) = fun_upd(A,option(B),F2,Xa,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_6305_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B3: A] :
      ( ( aa(B,option(A),M,A3) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M,A3,aa(A,option(A),some(A),B3))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),ran(B,A,M)) ) ) ).

% ran_map_upd
tff(fact_6306_nth__zip,axiom,
    ! [A: $tType,B: $tType,I: nat,Xs: list(A),Ys2: list(B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys2))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys2)),I) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Ys2),I)) ) ) ) ).

% nth_zip
tff(fact_6307_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_aby(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_6308_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_abz(list(A),fun(list(B),fun(product_prod(A,B),$o)),Xs),Ys2)) ).

% set_zip
tff(fact_6309_ran__map__upd__Some,axiom,
    ! [B: $tType,A: $tType,M: fun(B,option(A)),Xa: B,Ya: A,Z: A] :
      ( ( aa(B,option(A),M,Xa) = aa(A,option(A),some(A),Ya) )
     => ( inj_on(B,option(A),M,dom(B,A,M))
       => ( ~ member(A,Z,ran(B,A,M))
         => ( ran(B,A,fun_upd(B,option(A),M,Xa,aa(A,option(A),some(A),Z))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),ran(B,A,M)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z),bot_bot(set(A)))) ) ) ) ) ).

% ran_map_upd_Some
tff(fact_6310_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys2: list(B),I: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
     => ( distinct(A,Xs)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(B),nat,size_size(list(B)),Ys2))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys2)),aa(nat,A,nth(A,Xs),I)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys2),I)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_6311_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( ( dom(A,B,F2) = bot_bot(set(A)) )
    <=> ! [X: A] : aa(A,option(B),F2,X) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_6312_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_aca(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_6313_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),Xa: A,Ya: option(B)] :
      dom(A,B,fun_upd(A,option(B),F2,Xa,Ya)) = $ite(Ya = none(B),aa(set(A),set(A),minus_minus(set(A),dom(A,B,F2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),dom(A,B,F2))) ).

% dom_fun_upd
tff(fact_6314_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xs: list(A),Ys2: list(B)] : dom(A,B,map_upds(A,B,M,Xs,Ys2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys2),Xs))),dom(A,B,M)) ).

% dom_map_upds
tff(fact_6315_finite__ran,axiom,
    ! [B: $tType,A: $tType,P3: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,P3))
     => aa(set(B),$o,finite_finite2(B),ran(A,B,P3)) ) ).

% finite_ran
tff(fact_6316_insert__dom,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xa: B,Ya: A] :
      ( ( aa(B,option(A),F2,Xa) = aa(A,option(A),some(A),Ya) )
     => ( aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),dom(B,A,F2)) = dom(B,A,F2) ) ) ).

% insert_dom
tff(fact_6317_finite__dom__map__of,axiom,
    ! [B: $tType,A: $tType,L: list(product_prod(A,B))] : aa(set(A),$o,finite_finite2(A),dom(A,B,map_of(A,B,L))) ).

% finite_dom_map_of
tff(fact_6318_dom__if,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F2: fun(A,option(B)),G: fun(A,option(B))] : dom(A,B,aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_acb(fun(A,$o),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),P),F2),G)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,F2)),aa(fun(A,$o),set(A),collect(A),P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,G)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)))) ).

% dom_if
tff(fact_6319_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B))] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,F2))
     => ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
       => ? [X3: A] : aa(A,option(B),F2,X3) = none(B) ) ) ).

% finite_map_freshness
tff(fact_6320_dom__minus,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,option(A)),Xa: B,A4: set(B)] :
      ( ( aa(B,option(A),F2,Xa) = none(A) )
     => ( aa(set(B),set(B),minus_minus(set(B),dom(B,A,F2)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),A4)) = aa(set(B),set(B),minus_minus(set(B),dom(B,A,F2)),A4) ) ) ).

% dom_minus
tff(fact_6321_finite__set__of__finite__maps,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => aa(set(fun(A,option(B))),$o,finite_finite2(fun(A,option(B))),aa(fun(fun(A,option(B)),$o),set(fun(A,option(B))),collect(fun(A,option(B))),aa(set(B),fun(fun(A,option(B)),$o),aTP_Lamp_acc(set(A),fun(set(B),fun(fun(A,option(B)),$o)),A4),B2))) ) ) ).

% finite_set_of_finite_maps
tff(fact_6322_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(fun(A,option(B)),$o)] :
      ( aa(set(A),$o,finite_finite2(A),dom(A,B,M))
     => ( aa(fun(A,option(B)),$o,P,aTP_Lamp_aca(A,option(B)))
       => ( ! [K2: A,V3: B,M4: fun(A,option(B))] :
              ( aa(set(A),$o,finite_finite2(A),dom(A,B,M4))
             => ( ~ member(A,K2,dom(A,B,M4))
               => ( aa(fun(A,option(B)),$o,P,M4)
                 => aa(fun(A,option(B)),$o,P,fun_upd(A,option(B),M4,K2,aa(B,option(B),some(B),V3))) ) ) )
         => aa(fun(A,option(B)),$o,P,M) ) ) ) ).

% finite_Map_induct
tff(fact_6323_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,option(B)),Xa: A] :
      ( ( dom(A,B,F2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) )
    <=> ? [V6: B] : F2 = fun_upd(A,option(B),aTP_Lamp_aca(A,option(B)),Xa,aa(B,option(B),some(B),V6)) ) ).

% dom_eq_singleton_conv
tff(fact_6324_dom__override__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,option(B)),G: fun(A,option(B)),A4: set(A)] : dom(A,B,override_on(A,option(B),F2,G,A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),minus_minus(set(A),dom(A,B,F2)),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_acd(fun(A,option(B)),fun(set(A),fun(A,$o)),G),A4)))),aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ace(fun(A,option(B)),fun(set(A),fun(A,$o)),G),A4))) ).

% dom_override_on
tff(fact_6325_card__quotient__disjoint,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( inj_on(A,set(set(A)),aTP_Lamp_acf(set(product_prod(A,A)),fun(A,set(set(A))),R2),A4)
       => ( aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A4,R2)) = aa(set(A),nat,finite_card(A),A4) ) ) ) ).

% card_quotient_disjoint
tff(fact_6326_override__on__emptyset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B)] : override_on(A,B,F2,G,bot_bot(set(A))) = F2 ).

% override_on_emptyset
tff(fact_6327_override__on__insert_H,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B),Xa: A,X4: set(A)] : override_on(A,B,F2,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),X4)) = override_on(A,B,fun_upd(A,B,F2,Xa,aa(A,B,G,Xa)),G,X4) ).

% override_on_insert'
tff(fact_6328_override__on__insert,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),G: fun(A,B),Xa: A,X4: set(A)] : override_on(A,B,F2,G,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),X4)) = fun_upd(A,B,override_on(A,B,F2,G,X4),Xa,aa(A,B,G,Xa)) ).

% override_on_insert
tff(fact_6329_quotient__is__empty2,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( ( bot_bot(set(set(A))) = equiv_quotient(A,A4,R2) )
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% quotient_is_empty2
tff(fact_6330_quotient__is__empty,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( ( equiv_quotient(A,A4,R2) = bot_bot(set(set(A))) )
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% quotient_is_empty
tff(fact_6331_quotient__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : equiv_quotient(A,bot_bot(set(A)),R2) = bot_bot(set(set(A))) ).

% quotient_empty
tff(fact_6332_quotient__diff1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),A3: A] :
      ( inj_on(A,set(set(A)),aTP_Lamp_acf(set(product_prod(A,A)),fun(A,set(set(A))),R2),A4)
     => ( member(A,A3,A4)
       => ( equiv_quotient(A,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))),R2) = aa(set(set(A)),set(set(A)),minus_minus(set(set(A)),equiv_quotient(A,A4,R2)),equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))),R2)) ) ) ) ).

% quotient_diff1
tff(fact_6333_quotient__def,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] : equiv_quotient(A,A4,R2) = aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(A),set(set(set(A))),image2(A,set(set(A)),aTP_Lamp_acg(set(product_prod(A,A)),fun(A,set(set(A))),R2)),A4)) ).

% quotient_def
tff(fact_6334_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),Xa: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
         => ( aa(list(A),A,groups8242544230860333062m_list(A),list_update(A,Xs,K,Xa)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),Xa)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_6335_ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B)),A4: set(A)] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),R2)
     => ( member(A,A3,A4)
       => member(B,B3,aa(set(A),set(B),image(A,B,R2),A4)) ) ) ).

% ImageI
tff(fact_6336_Image__empty2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,A))] : aa(set(B),set(A),image(B,A,R),bot_bot(set(B))) = bot_bot(set(A)) ).

% Image_empty2
tff(fact_6337_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_6338_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) )
        <=> ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Ns))
             => ( X = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_6339_Image__empty1,axiom,
    ! [B: $tType,A: $tType,X4: set(B)] : aa(set(B),set(A),image(B,A,bot_bot(set(product_prod(B,A)))),X4) = bot_bot(set(A)) ).

% Image_empty1
tff(fact_6340_Image__Id__on,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : aa(set(A),set(A),image(A,A,id_on(A,A4)),B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) ).

% Image_Id_on
tff(fact_6341_Image__singleton__iff,axiom,
    ! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),A3: B] :
      ( member(A,B3,aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,A3),B3),R2) ) ).

% Image_singleton_iff
tff(fact_6342_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xa: A,Xs: list(A)] :
          ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).

% member_le_sum_list
tff(fact_6343_Image__Int__subset,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A4: set(B),B2: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B2))),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,R),A4)),aa(set(B),set(A),image(B,A,R),B2))) ).

% Image_Int_subset
tff(fact_6344_Image__UN,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),B2: fun(C,set(B)),A4: set(C)] : aa(set(B),set(A),image(B,A,R2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B2),A4))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ach(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B2)),A4)) ).

% Image_UN
tff(fact_6345_Image__closed__trancl,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),X4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),X4)),X4)
     => ( aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R2)),X4) = X4 ) ) ).

% Image_closed_trancl
tff(fact_6346_Image__mono,axiom,
    ! [B: $tType,A: $tType,R4: set(product_prod(A,B)),R2: set(product_prod(A,B)),A14: set(A),A4: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R4),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A14),A4)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,R4),A14)),aa(set(A),set(B),image(A,B,R2),A4)) ) ) ).

% Image_mono
tff(fact_6347_Un__Image,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),S: set(product_prod(B,A)),A4: set(B)] : aa(set(B),set(A),image(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),R),S)),A4) = 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,R),A4)),aa(set(B),set(A),image(B,A,S),A4)) ).

% Un_Image
tff(fact_6348_Image__Un,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A4: set(B),B2: set(B)] : aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B2)) = 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,R),A4)),aa(set(B),set(A),image(B,A,R),B2)) ).

% Image_Un
tff(fact_6349_rev__ImageI,axiom,
    ! [B: $tType,A: $tType,A3: A,A4: set(A),B3: B,R2: set(product_prod(A,B))] :
      ( member(A,A3,A4)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),R2)
       => member(B,B3,aa(set(A),set(B),image(A,B,R2),A4)) ) ) ).

% rev_ImageI
tff(fact_6350_Image__iff,axiom,
    ! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),A4: set(B)] :
      ( member(A,B3,aa(set(B),set(A),image(B,A,R2),A4))
    <=> ? [X: B] :
          ( member(B,X,A4)
          & member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,X),B3),R2) ) ) ).

% Image_iff
tff(fact_6351_ImageE,axiom,
    ! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A)),A4: set(B)] :
      ( member(A,B3,aa(set(B),set(A),image(B,A,R2),A4))
     => ~ ! [X3: B] :
            ( member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,X3),B3),R2)
           => ~ member(B,X3,A4) ) ) ).

% ImageE
tff(fact_6352_finite__Image,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),A4: set(A)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => aa(set(B),$o,finite_finite2(B),aa(set(A),set(B),image(A,B,R),A4)) ) ).

% finite_Image
tff(fact_6353_quotientE,axiom,
    ! [A: $tType,X4: set(A),A4: set(A),R2: set(product_prod(A,A))] :
      ( member(set(A),X4,equiv_quotient(A,A4,R2))
     => ~ ! [X3: A] :
            ( ( X4 = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A)))) )
           => ~ member(A,X3,A4) ) ) ).

% quotientE
tff(fact_6354_quotientI,axiom,
    ! [A: $tType,Xa: A,A4: set(A),R2: set(product_prod(A,A))] :
      ( member(A,Xa,A4)
     => member(set(A),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))),equiv_quotient(A,A4,R2)) ) ).

% quotientI
tff(fact_6355_finite__rtrancl__Image,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => aa(set(A),$o,finite_finite2(A),aa(set(A),set(A),image(A,A,transitive_rtrancl(A,R)),A4)) ) ) ).

% finite_rtrancl_Image
tff(fact_6356_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(list(A),A,groups8242544230860333062m_list(A),Xs)) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_6357_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),X3) )
         => ( ( aa(list(A),A,groups8242544230860333062m_list(A),Xs) = zero_zero(A) )
          <=> ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => ( X = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_6358_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),zero_zero(A)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs)),zero_zero(A)) ) ) ).

% sum_list_nonpos
tff(fact_6359_Image__singleton,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),A3: B] : aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B)))) = aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aTP_Lamp_aci(set(product_prod(B,A)),fun(B,fun(A,$o)),R2),A3)) ).

% Image_singleton
tff(fact_6360_Image__INT__subset,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),B2: fun(C,set(B)),A4: set(C)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B2),A4)))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ach(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B2)),A4))) ).

% Image_INT_subset
tff(fact_6361_Image__eq__UN,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),B2: set(B)] : aa(set(B),set(A),image(B,A,R2),B2) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_acj(set(product_prod(B,A)),fun(B,set(A)),R2)),B2)) ).

% Image_eq_UN
tff(fact_6362_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),aa(list(A),A,groups8242544230860333062m_list(A),Ns)) ) ) ).

% elem_le_sum_list
tff(fact_6363_UN__Image,axiom,
    ! [B: $tType,A: $tType,C: $tType,X4: fun(C,set(product_prod(B,A))),I5: set(C),S: set(B)] : aa(set(B),set(A),image(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),X4),I5))),S) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_ack(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),X4),S)),I5)) ).

% UN_Image
tff(fact_6364_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_acl(nat,fun(nat,fun(list(nat),$o)),M),N4))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_acm(nat,fun(nat,fun(list(nat),$o)),M),N4)))),aa(set(list(nat)),nat,finite_card(list(nat)),aa(fun(list(nat),$o),set(list(nat)),collect(list(nat)),aa(nat,fun(list(nat),$o),aTP_Lamp_acn(nat,fun(nat,fun(list(nat),$o)),M),N4)))) ) ) ).

% card_length_sum_list_rec
tff(fact_6365_singleton__quotient,axiom,
    ! [A: $tType,Xa: A,R2: set(product_prod(A,A))] : equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))),R2) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% singleton_quotient
tff(fact_6366_sum__list__sum__nth,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] : aa(list(A),A,groups8242544230860333062m_list(A),Xs) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_6367_less__eq__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_acp(nat,fun(nat,fun(product_prod(nat,nat),$o)))),rep_Integ(Xa)),rep_Integ(Xaa)) ) ).

% less_eq_int.rep_eq
tff(fact_6368_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X4: set(A),F2: fun(A,nat)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X4)
     => ( aa(set(A),$o,finite_finite2(A),X4)
       => ( 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_acq(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F2)),X4) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_6369_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_acr(B,A),Xs)) = zero_zero(A) ) ).

% sum_list_0
tff(fact_6370_nth__map,axiom,
    ! [B: $tType,A: $tType,Na: nat,Xs: list(A),F2: fun(A,B)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,map(A,B,F2,Xs)),Na) = aa(A,B,F2,aa(nat,A,nth(A,Xs),Na)) ) ) ).

% nth_map
tff(fact_6371_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xs: list(A),Ys2: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)))
     => ( ( map(A,B,F2,Xs) = map(A,B,F2,Ys2) )
      <=> ( Xs = Ys2 ) ) ) ).

% inj_on_map_eq_map
tff(fact_6372_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),Xs: list(B),Ys2: list(B)] :
      ( ( map(B,A,F2,Xs) = map(B,A,F2,Ys2) )
     => ( 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),Ys2)))
       => ( Xs = Ys2 ) ) ) ).

% map_inj_on
tff(fact_6373_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(list(A),A,groups8242544230860333062m_list(A),Xs))),aa(list(A),A,groups8242544230860333062m_list(A),map(A,A,abs_abs(A),Xs))) ) ).

% sum_list_abs
tff(fact_6374_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs))) ) ) ).

% sum_list_mono
tff(fact_6375_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F2: fun(A,B),G: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
               => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,Xs))),aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,G,Xs))) ) ) ) ).

% sum_list_strict_mono
tff(fact_6376_map__removeAll__inj__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Xa: A,Xs: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(list(A),set(A),set2(A),Xs)))
     => ( map(A,B,F2,removeAll(A,Xa,Xs)) = removeAll(B,aa(A,B,F2,Xa),map(A,B,F2,Xs)) ) ) ).

% map_removeAll_inj_on
tff(fact_6377_less__int_Orep__eq,axiom,
    ! [Xa: int,Xaa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Xa),Xaa)
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_act(nat,fun(nat,fun(product_prod(nat,nat),$o)))),rep_Integ(Xa)),rep_Integ(Xaa)) ) ).

% less_int.rep_eq
tff(fact_6378_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),Xb3: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),abs_Integ(Xa)),abs_Integ(Xb3))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_acp(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xa),Xb3) ) ).

% less_eq_int.abs_eq
tff(fact_6379_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),Xb3: product_prod(nat,nat)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),abs_Integ(Xa)),abs_Integ(Xb3))
    <=> aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_act(nat,fun(nat,fun(product_prod(nat,nat),$o)))),Xa),Xb3) ) ).

% less_int.abs_eq
tff(fact_6380_zero__int__def,axiom,
    zero_zero(int) = abs_Integ(aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),zero_zero(nat))) ).

% zero_int_def
tff(fact_6381_int__def,axiom,
    ! [Na: nat] : aa(nat,int,semiring_1_of_nat(int),Na) = abs_Integ(aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Na),zero_zero(nat))) ).

% int_def
tff(fact_6382_one__int__def,axiom,
    one_one(int) = abs_Integ(aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_6383_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,Na: nat] : aa(list($o),A,aa(A,fun(list($o),A),aa(fun($o,A),fun(A,fun(list($o),A)),groups4207007520872428315er_sum($o,A),zero_neq_one_of_bool(A)),aa(num,A,numeral_numeral(A),bit0(one2))),map(nat,$o,bit_se5641148757651400278ts_bit(A,A3),upt(zero_zero(nat),Na))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Na),A3) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_6384_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [F2: fun(nat,A),Ns: list(nat)] :
          ( ! [X3: nat,Y: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,X3)),aa(nat,A,F2,Y)) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),F2),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),aa(list(A),A,groups8242544230860333062m_list(A),map(nat,A,F2,Ns))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_6385_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( hd(nat,upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_6386_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_6387_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_6388_take__upt,axiom,
    ! [I: nat,M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)),Na)
     => ( take(nat,M,upt(I,Na)) = upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M)) ) ) ).

% take_upt
tff(fact_6389_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)
     => ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_6390_upt__rec__numeral,axiom,
    ! [M: num,Na: num] :
      upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),Na)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Na)),aa(list(nat),list(nat),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),Na))),nil(nat)) ).

% upt_rec_numeral
tff(fact_6391_sum__list__upt,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(list(nat),nat,groups8242544230860333062m_list(nat),upt(M,Na)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7311177749621191930dd_sum(nat,nat),aTP_Lamp_dt(nat,nat)),set_or7035219750837199246ssThan(nat,M,Na)) ) ) ).

% sum_list_upt
tff(fact_6392_map__add__upt,axiom,
    ! [Na: nat,M: nat] : map(nat,nat,aTP_Lamp_acu(nat,fun(nat,nat),Na),upt(zero_zero(nat),M)) = upt(Na,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Na)) ).

% map_add_upt
tff(fact_6393_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
        <=> sorted_wrt(B,aTP_Lamp_acv(fun(B,A),fun(B,fun(B,$o)),F2),Xs) ) ) ).

% sorted_map
tff(fact_6394_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = append(nat,upt(I,J),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_6395_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Na: nat,Xa: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,Na,Xa)) ) ).

% sorted_replicate
tff(fact_6396_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Na: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,Na,Xs)) ) ) ).

% sorted_drop
tff(fact_6397_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : sorted_wrt(A,ord_less(A),linord4507533701916653071of_set(A,A4)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_6398_sorted__wrt__upt,axiom,
    ! [M: nat,Na: nat] : sorted_wrt(nat,ord_less(nat),upt(M,Na)) ).

% sorted_wrt_upt
tff(fact_6399_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_6400_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : sorted_wrt(A,ord_less_eq(A),linord4507533701916653071of_set(A,A4)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_6401_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A3,Xs)) ) ) ).

% sorted_remove1
tff(fact_6402_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_6403_sorted__upt,axiom,
    ! [M: nat,Na: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,Na)) ).

% sorted_upt
tff(fact_6404_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_6405_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Na: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,Na,Xs)) ) ) ).

% sorted_take
tff(fact_6406_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_6407_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),linorder_insort_key(A,A,aTP_Lamp_jn(A,A),Xa,Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_6408_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ya: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xa),aa(list(A),list(A),cons(A,Ya),Zs)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Ya)
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Ya),Zs)) ) ) ) ).

% sorted2
tff(fact_6409_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_6410_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_6411_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_6412_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys2)
           => ( ( aa(list(A),set(A),set2(A),Ys2) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys2 = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_6413_atLeast__upt,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_lessThan(nat),Na) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Na)) ).

% atLeast_upt
tff(fact_6414_map__replicate__trivial,axiom,
    ! [A: $tType,Xa: A,I: nat] : map(nat,A,aTP_Lamp_acw(A,fun(nat,A),Xa),upt(zero_zero(nat),I)) = replicate(A,I,Xa) ).

% map_replicate_trivial
tff(fact_6415_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xa),nil(A))) ) ).

% sorted1
tff(fact_6416_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),cons(A,Xa),Ys2))
        <=> ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Ys2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X) )
            & sorted_wrt(A,ord_less_eq(A),Ys2) ) ) ) ).

% sorted_simps(2)
tff(fact_6417_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Ys2: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),cons(A,Xa),Ys2))
        <=> ( ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Ys2))
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),X) )
            & sorted_wrt(A,ord_less(A),Ys2) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_6418_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_6419_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),append(A,Xs,Ys2))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys2)
            & ! [X: A] :
                ( member(A,X,aa(list(A),set(A),set2(A),Xs))
               => ! [Xa3: A] :
                    ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys2))
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xa3) ) ) ) ) ) ).

% sorted_append
tff(fact_6420_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys2)
             => ( distinct(A,Ys2)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys2) )
                 => ( Xs = Ys2 ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_6421_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I4: nat,J3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_6422_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A),I: nat,J: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_6423_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
     => sorted_wrt(A,P,Xs) ) ).

% sorted_wrt01
tff(fact_6424_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xa: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,linorder_insort_key(B,A,F2,Xa,Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs)) ) ) ).

% sorted_insort_key
tff(fact_6425_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Xa: 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,Xa,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_6426_atMost__upto,axiom,
    ! [Na: nat] : aa(nat,set(nat),set_ord_atMost(nat),Na) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,Na))) ).

% atMost_upto
tff(fact_6427_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( upt(I,J) = aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6428_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J),aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_6429_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J),append(nat,upt(I,J),aa(list(nat),list(nat),cons(nat,J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_6430_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),aa(list(nat),list(nat),cons(nat,J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_6431_map__upt__Suc,axiom,
    ! [A: $tType,F2: fun(nat,A),Na: nat] : map(nat,A,F2,upt(zero_zero(nat),aa(nat,nat,suc,Na))) = aa(list(A),list(A),cons(A,aa(nat,A,F2,zero_zero(nat))),map(nat,A,aTP_Lamp_qt(fun(nat,A),fun(nat,A),F2),upt(zero_zero(nat),Na))) ).

% map_upt_Suc
tff(fact_6432_map__decr__upt,axiom,
    ! [M: nat,Na: nat] : map(nat,nat,aTP_Lamp_xw(nat,nat),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Na))) = upt(M,Na) ).

% map_decr_upt
tff(fact_6433_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_6434_sorted__iff__nth__mono__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono_less
tff(fact_6435_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat))
         => sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted01
tff(fact_6436_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ? [X3: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X3) = A4 )
              & sorted_wrt(A,ord_less_eq(A),X3)
              & distinct(A,X3)
              & ! [Y2: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y2) = A4 )
                    & sorted_wrt(A,ord_less_eq(A),Y2)
                    & distinct(A,Y2) )
                 => ( Y2 = X3 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_6437_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_6438_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Na: nat,M: nat,F2: fun(nat,A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(nat,nat,minus_minus(nat,Na),M))
     => ( aa(nat,A,nth(A,map(nat,A,F2,upt(M,Na))),I) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_6439_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,Xs: list(A)] :
          ( member(A,A3,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => ( linorder_insort_key(A,A,aTP_Lamp_jn(A,A),A3,remove1(A,A3,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_6440_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xa: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),cons(nat,Xa),Xs) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
        & ( I = Xa )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_6441_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))) ) ) ) ).

% sorted_iff_nth_Suc
tff(fact_6442_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ~ ! [L6: list(A)] :
                ( sorted_wrt(A,ord_less(A),L6)
               => ( ( aa(list(A),set(A),set2(A),L6) = A4 )
                 => ( aa(list(A),nat,size_size(list(A)),L6) != aa(set(A),nat,finite_card(A),A4) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_6443_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),J3)) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_6444_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),I)),aa(nat,A,nth(A,Xs),J)) ) ) ) ) ).

% sorted_nth_mono
tff(fact_6445_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(nat),nat,size_size(list(nat)),Ns))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,nth(nat,Ns),I)) ) ) ).

% sorted_wrt_less_idx
tff(fact_6446_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Na: nat,M: nat,F2: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,minus_minus(nat,Na),M) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( map(nat,A,F2,upt(M,Na)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_6447_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X2: A] :
                ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
                & aa(A,$o,P,X2) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_acx(list(A),fun(fun(A,$o),fun(A,$o)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_6448_map__sorted__distinct__set__unique,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xs: list(A),Ys2: list(A)] :
          ( inj_on(A,B,F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)))
         => ( sorted_wrt(B,ord_less_eq(B),map(A,B,F2,Xs))
           => ( distinct(B,map(A,B,F2,Xs))
             => ( sorted_wrt(B,ord_less_eq(B),map(A,B,F2,Ys2))
               => ( distinct(B,map(A,B,F2,Ys2))
                 => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys2) )
                   => ( Xs = Ys2 ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_6449_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),L: list(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ( sorted_wrt(A,ord_less(A),L)
              & ( aa(list(A),set(A),set2(A),L) = A4 )
              & ( aa(list(A),nat,size_size(list(A)),L) = aa(set(A),nat,finite_card(A),A4) ) )
          <=> ( linord4507533701916653071of_set(A,A4) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_6450_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A3: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),A3) )
           => ( linorder_insort_key(A,A,aTP_Lamp_jn(A,A),A3,Xs) = append(A,Xs,aa(list(A),list(A),cons(A,A3),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_6451_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),Na: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( Na = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = Na ) )
       => ( transpose(A,Xs) = map(nat,list(A),aTP_Lamp_acz(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),Na)) ) ) ) ).

% transpose_rectangle
tff(fact_6452_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ~ ! [L6: list(B)] :
                ( sorted_wrt(A,Less,map(B,A,F2,L6))
               => ( ( aa(list(B),set(B),set2(B),L6) = A4 )
                 => ( aa(list(B),nat,size_size(list(B)),L6) != aa(set(B),nat,finite_card(B),A4) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_6453_sorted__upto,axiom,
    ! [M: int,Na: int] : sorted_wrt(int,ord_less_eq(int),upto(M,Na)) ).

% sorted_upto
tff(fact_6454_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : sorted_wrt(int,ord_less(int),upto(I,J)) ).

% sorted_wrt_upto
tff(fact_6455_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_jn(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_6456_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( sorted_wrt(A,Less,map(B,A,F2,L))
              & ( aa(list(B),set(B),set2(B),L) = A4 )
              & ( aa(list(B),nat,size_size(list(B)),L) = aa(set(B),nat,finite_card(B),A4) ) )
          <=> ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A4) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_6457_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Xa: B,A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),A4)),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),minus_minus(set(B),A4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B))))) = remove1(B,Xa,sorted8670434370408473282of_set(A,B,Less_eq,F2,A4)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_6458_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B),B2: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),S)
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A4) = sorted8670434370408473282of_set(A,B,Less_eq,F2,B2) )
           => ( aa(set(B),$o,finite_finite2(B),A4)
             => ( aa(set(B),$o,finite_finite2(B),B2)
               => ( A4 = B2 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_6459_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_6460_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( aa(list(B),set(B),set2(B),sorted8670434370408473282of_set(A,B,Less_eq,F2,A4)) = A4 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_6461_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => ( aa(list(B),nat,size_size(list(B)),sorted8670434370408473282of_set(A,B,Less_eq,F2,A4)) = aa(set(B),nat,finite_card(B),A4) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_6462_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => distinct(A,map(B,A,F2,sorted8670434370408473282of_set(A,B,Less_eq,F2,A4))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_6463_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => sorted_wrt(A,Less_eq,map(B,A,F2,sorted8670434370408473282of_set(A,B,Less_eq,F2,A4))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_6464_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => sorted_wrt(A,Less,map(B,A,F2,sorted8670434370408473282of_set(A,B,Less_eq,F2,A4))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_6465_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ( sorted8670434370408473282of_set(A,B,Less_eq,F2,A4) = nil(B) )
          <=> ( A4 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_6466_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),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_6467_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Xa: B,A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),A4)),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),A4)) = insort_key(A,B,Less_eq,F2,Xa,sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),minus_minus(set(B),A4),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_6468_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),S: set(B),F2: fun(B,A),Xa: B,A4: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S,F2)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),A4)),S)
       => ( aa(set(B),$o,finite_finite2(B),A4)
         => ( ~ member(B,Xa,A4)
           => ( sorted8670434370408473282of_set(A,B,Less_eq,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),A4)) = insort_key(A,B,Less_eq,F2,Xa,sorted8670434370408473282of_set(A,B,Less_eq,F2,A4)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_6469_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = $ite(Xs = nil(list(A)),zero_zero(nat),aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat)))) ) ) ).

% length_transpose_sorted
tff(fact_6470_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_ada(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_6471_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ya: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( foldr(A,A,ord_max(A),Xs,Ya) = $ite(Xs = nil(A),Ya,aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,nth(A,Xs),zero_zero(nat))),Ya)) ) ) ) ).

% foldr_max_sorted
tff(fact_6472_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_6473_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_6474_rev__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,rev(A,Xs)),Na) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,Na))) ) ) ).

% rev_nth
tff(fact_6475_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ya: A] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs))
     => ( rev(A,list_update(A,Xs,K,Ya)) = list_update(A,rev(A,Xs),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Ya) ) ) ).

% rev_update
tff(fact_6476_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))),aa(nat,A,nth(A,Xs),I4)) ) ) ) ).

% sorted_rev_iff_nth_Suc
tff(fact_6477_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I)) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_6478_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,Xs))
        <=> ! [I4: nat,J3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),J3)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,nth(A,Xs),J3)),aa(nat,A,nth(A,Xs),I4)) ) ) ) ) ).

% sorted_rev_iff_nth_mono
tff(fact_6479_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F2: fun(B,A),A3: A,Xs: list(B)] : aa(list(B),A,aa(A,fun(list(B),A),aa(fun(B,A),fun(A,fun(list(B),A)),groups4207007520872428315er_sum(B,A),F2),A3),Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_adb(fun(B,A),fun(A,fun(B,fun(A,A))),F2),A3),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_6480_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_adc(nat,fun(list(A),$o),I),Xs)))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_6481_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( map(list(A),A,aTP_Lamp_add(nat,fun(list(A),A),I),filter2(list(A),aTP_Lamp_adc(nat,fun(list(A),$o),I),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I) ) ) ) ).

% transpose_column
tff(fact_6482_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_ade(fun(list(A),A),fun(list(A),fun(A,$o)),G),Xs),Xs)) ) ).

% sorted_same
tff(fact_6483_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_6484_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),filter2(A,P,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_6485_length__filter__less,axiom,
    ! [A: $tType,Xa: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xa,aa(list(A),set(A),set2(A),Xs))
     => ( ~ aa(A,$o,P,Xa)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_6486_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => sorted_wrt(A,ord_less_eq(A),map(B,A,F2,filter2(B,P,Xs))) ) ) ).

% sorted_filter
tff(fact_6487_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),Ya: A,Xs: list(A)] :
      ( inj_on(A,B,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),aa(list(A),set(A),set2(A),Xs)))
     => ( filter2(A,aa(A,fun(A,$o),aTP_Lamp_adf(fun(A,B),fun(A,fun(A,$o)),F2),Ya),Xs) = filter2(A,aa(A,fun(A,$o),fequal(A),Ya),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_6488_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),map(B,A,F2,filter2(B,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_adg(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),F2),G),Xs),Xs))) ) ).

% sorted_map_same
tff(fact_6489_sum__list__map__filter_H,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [F2: fun(B,A),P: fun(B,$o),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,$o),fun(B,A),aTP_Lamp_adh(fun(B,A),fun(fun(B,$o),fun(B,A)),F2),P),Xs)) ) ).

% sum_list_map_filter'
tff(fact_6490_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F2: fun(A,nat),P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,filter2(A,P,Xs)))),aa(list(nat),nat,groups8242544230860333062m_list(nat),map(A,nat,F2,Xs))) ).

% sum_list_filter_le_nat
tff(fact_6491_sum__list__map__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(B)
     => ! [Xs: list(A),P: fun(A,$o),F2: fun(A,B)] :
          ( ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
             => ( ~ aa(A,$o,P,X3)
               => ( aa(A,B,F2,X3) = zero_zero(B) ) ) )
         => ( aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,filter2(A,P,Xs))) = aa(list(B),B,groups8242544230860333062m_list(B),map(A,B,F2,Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_6492_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),P: fun(B,$o),Xa: B] :
          ( sorted_wrt(A,ord_less_eq(A),map(B,A,F2,Xs))
         => ( aa(B,$o,P,Xa)
           => ( filter2(B,P,linorder_insort_key(B,A,F2,Xa,Xs)) = linorder_insort_key(B,A,F2,Xa,filter2(B,P,Xs)) ) ) ) ) ).

% filter_insort
tff(fact_6493_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Ya: A] : aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aa(A,fun(A,$o),aTP_Lamp_adi(A,fun(A,$o)),Ya),Xs)) ).

% set_minus_filter_out
tff(fact_6494_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys2))
       => ( filter2(A,aTP_Lamp_adj(list(A),fun(A,$o),Ys2),Zs) = Ys2 ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_6495_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys2))
       => ( filter2(A,aTP_Lamp_adk(list(A),fun(A,$o),Ys2),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_6496_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys2))
       => ( filter2(A,aTP_Lamp_adj(list(A),fun(A,$o),Xs),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_6497_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys2: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys2)) = bot_bot(set(A)) )
     => ( member(list(A),Zs,shuffles(A,Xs,Ys2))
       => ( filter2(A,aTP_Lamp_adk(list(A),fun(A,$o),Xs),Zs) = Ys2 ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_6498_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : filter2(A,P,Xs) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_adl(fun(A,$o),fun(list(A),fun(nat,$o)),P),Xs))) ).

% filter_eq_nths
tff(fact_6499_length__filter__conv__card,axiom,
    ! [A: $tType,P3: fun(A,$o),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),filter2(A,P3,Xs)) = aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(list(A),fun(nat,$o),aTP_Lamp_adl(fun(A,$o),fun(list(A),fun(nat,$o)),P3),Xs))) ).

% length_filter_conv_card
tff(fact_6500_insort__key__remove1,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [A3: A,Xs: list(A),F2: fun(A,B)] :
          ( member(A,A3,aa(list(A),set(A),set2(A),Xs))
         => ( sorted_wrt(B,ord_less_eq(B),map(A,B,F2,Xs))
           => ( ( hd(A,filter2(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_adm(A,fun(fun(A,B),fun(A,$o)),A3),F2),Xs)) = A3 )
             => ( linorder_insort_key(A,B,F2,A3,remove1(A,A3,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_6501_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),foldr(list(B),nat,aTP_Lamp_adn(list(B),fun(nat,nat)),Xss,zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(list(A),nat,size_size(list(A)),Xs)),foldr(list(B),nat,aTP_Lamp_ado(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_adp(list(B),$o),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_6502_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = map(list(A),A,aTP_Lamp_add(nat,fun(list(A),A),I),filter2(list(A),aTP_Lamp_adc(nat,fun(list(A),$o),I),Xs)) ) ) ).

% nth_transpose
tff(fact_6503_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_ada(list(A),fun(nat,nat)),transpose(A,Xs),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_adq(list(A),$o),Xs)) ).

% transpose_max_length
tff(fact_6504_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_adc(nat,fun(list(A),$o),I),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I)) ) ) ) ).

% transpose_column_length
tff(fact_6505_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),rev(nat,map(list(A),nat,size_size(list(A)),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_adq(list(A),$o),Xs) ) ) ).

% transpose_transpose
tff(fact_6506_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Ta: A] :
          ( sorted_wrt(A,ord_less_eq(A),rev(A,map(B,A,F2,Xs)))
         => ( filter2(B,aa(A,fun(B,$o),aTP_Lamp_adr(fun(B,A),fun(A,fun(B,$o)),F2),Ta),Xs) = takeWhile(B,aa(A,fun(B,$o),aTP_Lamp_adr(fun(B,A),fun(A,fun(B,$o)),F2),Ta),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_6507_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_takeWhile_le
tff(fact_6508_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_6509_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))
     => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ).

% nth_length_takeWhile
tff(fact_6510_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,takeWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% takeWhile_nth
tff(fact_6511_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),J)
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_6512_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,Na: nat,Xs: list(A),P: fun(A,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),Na)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) ) )
     => ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
         => ~ aa(A,$o,P,aa(nat,A,nth(A,Xs),Na)) )
       => ( takeWhile(A,P,Xs) = take(A,Na,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_6513_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image2(B,A,F2),aa(list(B),set(B),set2(B),Xs))) = fold(B,A,aa(fun(B,A),fun(B,fun(A,A)),comp(A,fun(A,A),B,sup_sup(A)),F2),Xs,bot_bot(A)) ) ).

% SUP_set_fold
tff(fact_6514_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X4: set(A)] :
      ( countable_countable(A,X4)
     => ~ ! [F4: fun(nat,set(A))] :
            ( ! [I3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F4,I3)),X4)
           => ( ! [I3: nat] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),F4,I3)),aa(nat,set(A),F4,aa(nat,nat,suc,I3)))
             => ( ! [I3: nat] : aa(set(A),$o,finite_finite2(A),aa(nat,set(A),F4,I3))
               => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),F4),top_top(set(nat)))) != X4 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_6515_countable__empty,axiom,
    ! [A: $tType] : countable_countable(A,bot_bot(set(A))) ).

% countable_empty
tff(fact_6516_countable__insert__eq,axiom,
    ! [A: $tType,Xa: A,A4: set(A)] :
      ( countable_countable(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4))
    <=> countable_countable(A,A4) ) ).

% countable_insert_eq
tff(fact_6517_countable__insert,axiom,
    ! [A: $tType,A4: set(A),A3: A] :
      ( countable_countable(A,A4)
     => countable_countable(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) ) ).

% countable_insert
tff(fact_6518_countable__Un__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
    <=> ( countable_countable(A,A4)
        & countable_countable(A,B2) ) ) ).

% countable_Un_iff
tff(fact_6519_countable__Un,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( countable_countable(A,A4)
     => ( countable_countable(A,B2)
       => countable_countable(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) ) ) ).

% countable_Un
tff(fact_6520_ccSup__insert,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),A3: A] :
          ( countable_countable(A,A4)
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ).

% ccSup_insert
tff(fact_6521_ccInf__insert,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),A3: A] :
          ( countable_countable(A,A4)
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ) ) ).

% ccInf_insert
tff(fact_6522_countable__Diff__eq,axiom,
    ! [A: $tType,A4: set(A),Xa: A] :
      ( countable_countable(A,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))
    <=> countable_countable(A,A4) ) ).

% countable_Diff_eq
tff(fact_6523_ccSUP__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),F2: fun(A,B),A3: A] :
          ( countable_countable(A,A4)
         => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F2,A3)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ).

% ccSUP_insert
tff(fact_6524_ccINF__insert,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),F2: fun(A,B),A3: A] :
          ( countable_countable(A,A4)
         => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,F2,A3)),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ).

% ccINF_insert
tff(fact_6525_countable__image__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image2(B,A,F2),S))
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S)
          & ( aa(set(B),set(A),image2(B,A,F2),S) = aa(set(B),set(A),image2(B,A,F2),T9) ) ) ) ).

% countable_image_eq
tff(fact_6526_countable__subset__image,axiom,
    ! [A: $tType,B: $tType,B2: set(A),F2: fun(B,A),A4: set(B)] :
      ( ( countable_countable(A,B2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),A4)) )
    <=> ? [A17: set(B)] :
          ( countable_countable(B,A17)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A17),A4)
          & ( B2 = aa(set(B),set(A),image2(B,A,F2),A17) ) ) ) ).

% countable_subset_image
tff(fact_6527_ex__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),$o)] :
      ( ? [T9: set(A)] :
          ( countable_countable(A,T9)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S))
          & aa(set(A),$o,P,T9) )
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S)
          & aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% ex_countable_subset_image
tff(fact_6528_all__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),$o)] :
      ( ! [T9: set(A)] :
          ( ( countable_countable(A,T9)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S)) )
         => aa(set(A),$o,P,T9) )
    <=> ! [T9: set(B)] :
          ( ( countable_countable(B,T9)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S) )
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% all_countable_subset_image
tff(fact_6529_ccInf__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B2: set(A),A4: set(A)] :
          ( countable_countable(A,B2)
         => ( countable_countable(A,A4)
           => ( ! [B5: A] :
                  ( member(A,B5,B2)
                 => ? [X2: A] :
                      ( member(A,X2,A4)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),B5) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ) ) ).

% ccInf_mono
tff(fact_6530_ccInf__lower,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),Xa: A] :
          ( countable_countable(A,A4)
         => ( member(A,Xa,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),Xa) ) ) ) ).

% ccInf_lower
tff(fact_6531_ccInf__lower2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),U: A,V2: A] :
          ( countable_countable(A,A4)
         => ( member(A,U,A4)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),V2) ) ) ) ) ).

% ccInf_lower2
tff(fact_6532_le__ccInf__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),B3: A] :
          ( countable_countable(A,A4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),aa(set(A),A,complete_Inf_Inf(A),A4))
          <=> ! [X: A] :
                ( member(A,X,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),X) ) ) ) ) ).

% le_ccInf_iff
tff(fact_6533_ccInf__greatest,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),Z: A] :
          ( countable_countable(A,A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),X3) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A4)) ) ) ) ).

% ccInf_greatest
tff(fact_6534_countable__subset,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( countable_countable(A,B2)
       => countable_countable(A,A4) ) ) ).

% countable_subset
tff(fact_6535_ccSup__upper2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),U: A,V2: A] :
          ( countable_countable(A,A4)
         => ( member(A,U,A4)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),U)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),V2),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ) ).

% ccSup_upper2
tff(fact_6536_ccSup__le__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),B3: A] :
          ( countable_countable(A,A4)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),B3)
          <=> ! [X: A] :
                ( member(A,X,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),B3) ) ) ) ) ).

% ccSup_le_iff
tff(fact_6537_ccSup__upper,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),Xa: A] :
          ( countable_countable(A,A4)
         => ( member(A,Xa,A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),aa(set(A),A,complete_Sup_Sup(A),A4)) ) ) ) ).

% ccSup_upper
tff(fact_6538_ccSup__least,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),Z: A] :
          ( countable_countable(A,A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),Z) ) ) ) ).

% ccSup_least
tff(fact_6539_ccSup__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B2: set(A),A4: set(A)] :
          ( countable_countable(A,B2)
         => ( countable_countable(A,A4)
           => ( ! [A5: A] :
                  ( member(A,A5,A4)
                 => ? [X2: A] :
                      ( member(A,X2,B2)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A5),X2) ) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ) ) ).

% ccSup_mono
tff(fact_6540_less__ccSup__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S: set(A),A3: A] :
          ( countable_countable(A,S)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),aa(set(A),A,complete_Sup_Sup(A),S))
          <=> ? [X: A] :
                ( member(A,X,S)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),A3),X) ) ) ) ) ).

% less_ccSup_iff
tff(fact_6541_ccInf__less__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S: set(A),A3: A] :
          ( countable_countable(A,S)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S)),A3)
          <=> ? [X: A] :
                ( member(A,X,S)
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),A3) ) ) ) ) ).

% ccInf_less_iff
tff(fact_6542_to__nat__on__surj,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( countable_countable(A,A4)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ? [X3: A] :
            ( member(A,X3,A4)
            & ( aa(A,nat,countable_to_nat_on(A,A4),X3) = Na ) ) ) ) ).

% to_nat_on_surj
tff(fact_6543_countable__finite,axiom,
    ! [A: $tType,S: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => countable_countable(A,S) ) ).

% countable_finite
tff(fact_6544_uncountable__infinite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ countable_countable(A,A4)
     => ~ aa(set(A),$o,finite_finite2(A),A4) ) ).

% uncountable_infinite
tff(fact_6545_countable__Collect__finite,axiom,
    ! [A: $tType] :
      ( countable(A)
     => countable_countable(set(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A))) ) ).

% countable_Collect_finite
tff(fact_6546_infinite__countable__subset_H,axiom,
    ! [A: $tType,X4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),X4)
     => ? [C7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C7),X4)
          & countable_countable(A,C7)
          & ~ aa(set(A),$o,finite_finite2(A),C7) ) ) ).

% infinite_countable_subset'
tff(fact_6547_countable__Collect__finite__subset,axiom,
    ! [A: $tType,T3: set(A)] :
      ( countable_countable(A,T3)
     => countable_countable(set(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aaz(set(A),fun(set(A),$o),T3))) ) ).

% countable_Collect_finite_subset
tff(fact_6548_ccSup__subset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B2: set(A),A4: set(A)] :
          ( countable_countable(A,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ) ).

% ccSup_subset_mono
tff(fact_6549_ccInf__superset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( countable_countable(A,A4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ) ).

% ccInf_superset_mono
tff(fact_6550_countable__image__eq__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image2(B,A,F2),S))
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S)
          & ( aa(set(B),set(A),image2(B,A,F2),S) = aa(set(B),set(A),image2(B,A,F2),T9) )
          & inj_on(B,A,F2,T9) ) ) ).

% countable_image_eq_inj
tff(fact_6551_ex__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),$o)] :
      ( ? [T9: set(A)] :
          ( countable_countable(A,T9)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S))
          & aa(set(A),$o,P,T9) )
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S)
          & inj_on(B,A,F2,T9)
          & aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% ex_countable_subset_image_inj
tff(fact_6552_all__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),S: set(B),P: fun(set(A),$o)] :
      ( ! [T9: set(A)] :
          ( ( countable_countable(A,T9)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T9),aa(set(B),set(A),image2(B,A,F2),S)) )
         => aa(set(A),$o,P,T9) )
    <=> ! [T9: set(B)] :
          ( ( countable_countable(B,T9)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T9),S)
            & inj_on(B,A,F2,T9) )
         => aa(set(A),$o,P,aa(set(B),set(A),image2(B,A,F2),T9)) ) ) ).

% all_countable_subset_image_inj
tff(fact_6553_ccSup__union__distrib,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( countable_countable(A,A4)
         => ( countable_countable(A,B2)
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2)) ) ) ) ) ).

% ccSup_union_distrib
tff(fact_6554_ccInf__union__distrib,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( countable_countable(A,A4)
         => ( countable_countable(A,B2)
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2)) ) ) ) ) ).

% ccInf_union_distrib
tff(fact_6555_countable__Image,axiom,
    ! [B: $tType,A: $tType,Y3: set(A),X4: set(product_prod(A,B))] :
      ( ! [Y: A] :
          ( member(A,Y,Y3)
         => countable_countable(B,aa(set(A),set(B),image(A,B,X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))) )
     => ( countable_countable(A,Y3)
       => countable_countable(B,aa(set(A),set(B),image(A,B,X4),Y3)) ) ) ).

% countable_Image
tff(fact_6556_uncountable__def,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ countable_countable(A,A4)
    <=> ( ( A4 != bot_bot(set(A)) )
        & ~ ? [F7: fun(nat,A)] : aa(set(nat),set(A),image2(nat,A,F7),top_top(set(nat))) = A4 ) ) ).

% uncountable_def
tff(fact_6557_union__set__fold,axiom,
    ! [A: $tType,Xs: list(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),A4) = fold(A,set(A),insert2(A),Xs,A4) ).

% union_set_fold
tff(fact_6558_ccSUP__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( counta3822494911875563373attice(C)
     => ! [A4: set(A),B2: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( countable_countable(A,A4)
         => ( countable_countable(B,B2)
           => ( ! [N2: A] :
                  ( member(A,N2,A4)
                 => ? [X2: B] :
                      ( member(B,X2,B2)
                      & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,N2)),aa(B,C,G,X2)) ) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Sup_Sup(C),aa(set(A),set(C),image2(A,C,F2),A4))),aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,G),B2))) ) ) ) ) ).

% ccSUP_mono
tff(fact_6559_ccSUP__least,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),F2: fun(A,B),U: B] :
          ( countable_countable(A,A4)
         => ( ! [I2: A] :
                ( member(A,I2,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I2)),U) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),U) ) ) ) ).

% ccSUP_least
tff(fact_6560_ccSUP__upper,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),I: A,F2: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( member(A,I,A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ).

% ccSUP_upper
tff(fact_6561_ccSUP__le__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),F2: fun(A,B),U: B] :
          ( countable_countable(A,A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),U)
          <=> ! [X: A] :
                ( member(A,X,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X)),U) ) ) ) ) ).

% ccSUP_le_iff
tff(fact_6562_ccSUP__upper2,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),I: A,U: B,F2: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( member(A,I,A4)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I))
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ) ).

% ccSUP_upper2
tff(fact_6563_countable__infiniteE_H,axiom,
    ! [A: $tType,A4: set(A)] :
      ( countable_countable(A,A4)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ~ ! [G7: fun(nat,A)] : ~ bij_betw(nat,A,G7,top_top(set(nat)),A4) ) ) ).

% countable_infiniteE'
tff(fact_6564_less__ccSUP__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(B)
        & linorder(B) )
     => ! [A4: set(A),A3: B,F2: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4)))
          <=> ? [X: A] :
                ( member(A,X,A4)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),A3),aa(A,B,F2,X)) ) ) ) ) ).

% less_ccSUP_iff
tff(fact_6565_ccINF__greatest,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),U: B,F2: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( ! [I2: A] :
                ( member(A,I2,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,I2)) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ).

% ccINF_greatest
tff(fact_6566_le__ccINF__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),U: B,F2: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4)))
          <=> ! [X: A] :
                ( member(A,X,A4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),U),aa(A,B,F2,X)) ) ) ) ) ).

% le_ccINF_iff
tff(fact_6567_ccINF__lower2,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),I: A,F2: fun(A,B),U: B] :
          ( countable_countable(A,A4)
         => ( member(A,I,A4)
           => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,I)),U)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),U) ) ) ) ) ).

% ccINF_lower2
tff(fact_6568_ccINF__lower,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),I: A,F2: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( member(A,I,A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(A,B,F2,I)) ) ) ) ).

% ccINF_lower
tff(fact_6569_ccINF__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( counta3822494911875563373attice(C)
     => ! [A4: set(A),B2: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( countable_countable(A,A4)
         => ( countable_countable(B,B2)
           => ( ! [M4: B] :
                  ( member(B,M4,B2)
                 => ? [X2: A] :
                      ( member(A,X2,A4)
                      & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F2,X2)),aa(B,C,G,M4)) ) )
             => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,F2),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),B2))) ) ) ) ) ).

% ccINF_mono
tff(fact_6570_ccINF__less__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(B)
        & linorder(B) )
     => ! [A4: set(A),F2: fun(A,B),A3: B] :
          ( countable_countable(A,A4)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),A3)
          <=> ? [X: A] :
                ( member(A,X,A4)
                & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,X)),A3) ) ) ) ) ).

% ccINF_less_iff
tff(fact_6571_ccSUP__sup__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),A4))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ads(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)),A4)) ) ) ) ).

% ccSUP_sup_distrib
tff(fact_6572_ccINF__sup__distrib2,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( counta4013691401010221786attice(C)
     => ! [A4: set(A),B2: set(B),F2: fun(A,C),G: fun(B,C)] :
          ( countable_countable(A,A4)
         => ( countable_countable(B,B2)
           => ( aa(C,C,aa(C,fun(C,C),sup_sup(C),aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,F2),A4))),aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,G),B2))) = aa(set(C),C,complete_Inf_Inf(C),aa(set(A),set(C),image2(A,C,aa(fun(B,C),fun(A,C),aa(fun(A,C),fun(fun(B,C),fun(A,C)),aTP_Lamp_adu(set(B),fun(fun(A,C),fun(fun(B,C),fun(A,C))),B2),F2),G)),A4)) ) ) ) ) ).

% ccINF_sup_distrib2
tff(fact_6573_sup__ccInf,axiom,
    ! [A: $tType] :
      ( counta4013691401010221786attice(A)
     => ! [B2: set(A),A3: A] :
          ( countable_countable(A,B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),aa(set(A),A,complete_Inf_Inf(A),B2)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aa(A,fun(A,A),sup_sup(A),A3)),B2)) ) ) ) ).

% sup_ccInf
tff(fact_6574_sup__ccINF,axiom,
    ! [B: $tType,A: $tType] :
      ( counta4013691401010221786attice(B)
     => ! [B2: set(A),A3: B,F2: fun(A,B)] :
          ( countable_countable(A,B2)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),A3),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),B2))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adv(B,fun(fun(A,B),fun(A,B)),A3),F2)),B2)) ) ) ) ).

% sup_ccINF
tff(fact_6575_ccInf__sup,axiom,
    ! [A: $tType] :
      ( counta4013691401010221786attice(A)
     => ! [B2: set(A),A3: A] :
          ( countable_countable(A,B2)
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),B2)),A3) = aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),image2(A,A,aTP_Lamp_adw(A,fun(A,A),A3)),B2)) ) ) ) ).

% ccInf_sup
tff(fact_6576_ccINF__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( counta4013691401010221786attice(B)
     => ! [B2: set(A),F2: fun(A,B),A3: B] :
          ( countable_countable(A,B2)
         => ( aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),B2))),A3) = aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,aa(B,fun(A,B),aTP_Lamp_adx(fun(A,B),fun(B,fun(A,B)),F2),A3)),B2)) ) ) ) ).

% ccINF_sup
tff(fact_6577_countableE__infinite,axiom,
    ! [A: $tType,S: set(A)] :
      ( countable_countable(A,S)
     => ( ~ aa(set(A),$o,finite_finite2(A),S)
       => ~ ! [E: fun(A,nat)] : ~ bij_betw(A,nat,E,S,top_top(set(nat))) ) ) ).

% countableE_infinite
tff(fact_6578_ccSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( countable_countable(A,A4)
         => ( countable_countable(A,B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A4)),aa(set(A),A,complete_Sup_Sup(A),B2))) ) ) ) ).

% ccSup_inter_less_eq
tff(fact_6579_less__eq__ccInf__inter,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A4: set(A),B2: set(A)] :
          ( countable_countable(A,A4)
         => ( countable_countable(A,B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A4)),aa(set(A),A,complete_Inf_Inf(A),B2))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))) ) ) ) ).

% less_eq_ccInf_inter
tff(fact_6580_ccSUP__subset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [B2: set(A),A4: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( countable_countable(A,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
           => ( ! [X3: A] :
                  ( member(A,X3,A4)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,G),B2))) ) ) ) ) ).

% ccSUP_subset_mono
tff(fact_6581_ccINF__superset__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),B2: set(A),F2: fun(A,B),G: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
           => ( ! [X3: A] :
                  ( member(A,X3,B2)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,G),B2))) ) ) ) ) ).

% ccINF_superset_mono
tff(fact_6582_Sup__set__fold,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Xs: list(A)] : aa(set(A),A,complete_Sup_Sup(A),aa(list(A),set(A),set2(A),Xs)) = fold(A,A,sup_sup(A),Xs,bot_bot(A)) ) ).

% Sup_set_fold
tff(fact_6583_ccSUP__union,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),B2: set(A),M5: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( countable_countable(A,B2)
           => ( aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,M5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,M5),A4))),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,M5),B2))) ) ) ) ) ).

% ccSUP_union
tff(fact_6584_mono__ccSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( countable_countable(A,A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image2(A,B,F2),A4))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),A4))) ) ) ) ).

% mono_ccSup
tff(fact_6585_mono__ccSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),I5: set(C),A4: fun(C,A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( countable_countable(C,I5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ady(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4)),I5))),aa(A,B,F2,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image2(C,A,A4),I5)))) ) ) ) ).

% mono_ccSUP
tff(fact_6586_ccINF__union,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [A4: set(A),B2: set(A),M5: fun(A,B)] :
          ( countable_countable(A,A4)
         => ( countable_countable(A,B2)
           => ( aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,M5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,M5),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,M5),B2))) ) ) ) ) ).

% ccINF_union
tff(fact_6587_mono__ccINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta4013691401010221786attice(A) )
     => ! [F2: fun(A,B),I5: set(C),A4: fun(C,A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( countable_countable(C,I5)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,A4),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image2(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ady(fun(A,B),fun(fun(C,A),fun(C,B)),F2),A4)),I5))) ) ) ) ).

% mono_ccINF
tff(fact_6588_mono__ccInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),A4: set(A)] :
          ( aa(fun(A,B),$o,order_mono(A,B),F2)
         => ( countable_countable(A,A4)
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,aa(set(A),A,complete_Inf_Inf(A),A4))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image2(A,B,F2),A4))) ) ) ) ).

% mono_ccInf
tff(fact_6589_countable__as__injective__image,axiom,
    ! [A: $tType,A4: set(A)] :
      ( countable_countable(A,A4)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ~ ! [F5: fun(nat,A)] :
              ( ( A4 = aa(set(nat),set(A),image2(nat,A,F5),top_top(set(nat))) )
             => ~ inj_on(nat,A,F5,top_top(set(nat))) ) ) ) ).

% countable_as_injective_image
tff(fact_6590_image__to__nat__on,axiom,
    ! [A: $tType,A4: set(A)] :
      ( countable_countable(A,A4)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),set(nat),image2(A,nat,countable_to_nat_on(A,A4)),A4) = top_top(set(nat)) ) ) ) ).

% image_to_nat_on
tff(fact_6591_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Xa: A,Xs: list(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),cons(A,Xa),Xs))) = fold(A,A,sup_sup(A),Xs,Xa) ) ).

% Sup_fin.set_eq_fold
tff(fact_6592_to__nat__on__infinite,axiom,
    ! [A: $tType,S: set(A)] :
      ( countable_countable(A,S)
     => ( ~ aa(set(A),$o,finite_finite2(A),S)
       => bij_betw(A,nat,countable_to_nat_on(A,S),S,top_top(set(nat))) ) ) ).

% to_nat_on_infinite
tff(fact_6593_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Ya: B] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S)
       => ( finite_fold(A,B,F2,Ya,aa(list(A),set(A),set2(A),Xs)) = fold(A,B,F2,Xs,Ya) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_6594_countable__enum__cases,axiom,
    ! [A: $tType,S: set(A)] :
      ( countable_countable(A,S)
     => ( ( aa(set(A),$o,finite_finite2(A),S)
         => ! [F5: fun(A,nat)] : ~ bij_betw(A,nat,F5,S,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),S))) )
       => ~ ( ~ aa(set(A),$o,finite_finite2(A),S)
           => ! [F5: fun(A,nat)] : ~ bij_betw(A,nat,F5,S,top_top(set(nat))) ) ) ) ).

% countable_enum_cases
tff(fact_6595_range__from__nat__into,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( countable_countable(A,A4)
       => ( aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A4)),top_top(set(nat))) = A4 ) ) ) ).

% range_from_nat_into
tff(fact_6596_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xs: list(A),Ya: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S)
       => ( finite_fold(A,B,F2,Ya,aa(list(A),set(A),set2(A),Xs)) = fold(A,B,F2,remdups(A,Xs),Ya) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_6597_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_remdups_leq
tff(fact_6598_from__nat__into__inject,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( countable_countable(A,A4)
       => ( ( B2 != bot_bot(set(A)) )
         => ( countable_countable(A,B2)
           => ( ( aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A4) = aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),B2) )
            <=> ( A4 = B2 ) ) ) ) ) ) ).

% from_nat_into_inject
tff(fact_6599_from__nat__into__inj__infinite,axiom,
    ! [A: $tType,A4: set(A),M: nat,Na: nat] :
      ( countable_countable(A,A4)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ( ( aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A4),M) = aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A4),Na) )
        <=> ( M = Na ) ) ) ) ).

% from_nat_into_inj_infinite
tff(fact_6600_to__nat__on__from__nat__into__infinite,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( countable_countable(A,A4)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(A,nat,countable_to_nat_on(A,A4),aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A4),Na)) = Na ) ) ) ).

% to_nat_on_from_nat_into_infinite
tff(fact_6601_from__nat__into,axiom,
    ! [A: $tType,A4: set(A),Na: nat] :
      ( ( A4 != bot_bot(set(A)) )
     => member(A,aa(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A4),Na),A4) ) ).

% from_nat_into
tff(fact_6602_sorted__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups(A,Xs)) ) ) ).

% sorted_remdups
tff(fact_6603_inj__on__from__nat__into,axiom,
    ! [A: $tType] : inj_on(set(A),fun(nat,A),counta4804993851260445106t_into(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_adz(set(A),$o))) ).

% inj_on_from_nat_into
tff(fact_6604_range__from__nat__into__subset,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ( A4 != bot_bot(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A4)),top_top(set(nat)))),A4) ) ).

% range_from_nat_into_subset
tff(fact_6605_subset__range__from__nat__into,axiom,
    ! [A: $tType,A4: set(A)] :
      ( countable_countable(A,A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(nat),set(A),image2(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),A4)),top_top(set(nat)))) ) ).

% subset_range_from_nat_into
tff(fact_6606_bij__betw__from__nat__into__finite,axiom,
    ! [A: $tType,S: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),S)
     => bij_betw(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),S),aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),S)),S) ) ).

% bij_betw_from_nat_into_finite
tff(fact_6607_bij__betw__from__nat__into,axiom,
    ! [A: $tType,S: set(A)] :
      ( countable_countable(A,S)
     => ( ~ aa(set(A),$o,finite_finite2(A),S)
       => bij_betw(nat,A,aa(set(A),fun(nat,A),counta4804993851260445106t_into(A),S),top_top(set(nat)),S) ) ) ).

% bij_betw_from_nat_into
tff(fact_6608_comp__fun__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S,F2)
    <=> ( finite4664212375090638736ute_on(A,B,S,F2)
        & finite4980608107308702382axioms(A,B,S,F2) ) ) ).

% comp_fun_idem_on_def
tff(fact_6609_comp__fun__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( finite4980608107308702382axioms(A,B,S,F2)
       => finite673082921795544331dem_on(A,B,S,F2) ) ) ).

% comp_fun_idem_on.intro
tff(fact_6610_comp__fun__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite4980608107308702382axioms(A,B,S,F2)
    <=> ! [X: A] :
          ( member(A,X,S)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,X)) = aa(A,fun(B,B),F2,X) ) ) ) ).

% comp_fun_idem_on_axioms_def
tff(fact_6611_comp__fun__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A] :
          ( member(A,X3,S)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,X3)) = aa(A,fun(B,B),F2,X3) ) )
     => finite4980608107308702382axioms(A,B,S,F2) ) ).

% comp_fun_idem_on_axioms.intro
tff(fact_6612_comp__fun__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite673082921795544331dem_on(A,B,S,F2)
     => finite4980608107308702382axioms(A,B,S,F2) ) ).

% comp_fun_idem_on.axioms(2)
tff(fact_6613_Field__insert,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3)),R2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ).

% Field_insert
tff(fact_6614_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),Z: B,Ya: B,A3: A] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),Ya)
         => ( member(A,A3,A4)
           => ? [Y6: B] :
                ( ( Ya = aa(B,B,aa(A,fun(B,B),F2,A3),Y6) )
                & aa(B,$o,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),Y6) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_6615_Field__empty,axiom,
    ! [A: $tType] : aa(set(product_prod(A,A)),set(A),field2(A),bot_bot(set(product_prod(A,A)))) = bot_bot(set(A)) ).

% Field_empty
tff(fact_6616_Field__Un,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S3)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),R2)),aa(set(product_prod(A,A)),set(A),field2(A),S3)) ).

% Field_Un
tff(fact_6617_Field__Union,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] : aa(set(product_prod(A,A)),set(A),field2(A),aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,A))),set(set(A)),image2(set(product_prod(A,A)),set(A),field2(A)),R)) ).

% Field_Union
tff(fact_6618_fold__graph_Osimps,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,A12: set(A),A23: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A12),A23)
    <=> ( ( ( A12 = bot_bot(set(A)) )
          & ( A23 = Z ) )
        | ? [X: A,A8: set(A),Y4: B] :
            ( ( A12 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),A8) )
            & ( A23 = aa(B,B,aa(A,fun(B,B),F2,X),Y4) )
            & ~ member(A,X,A8)
            & aa(B,$o,finite_fold_graph(A,B,F2,Z,A8),Y4) ) ) ) ).

% fold_graph.simps
tff(fact_6619_fold__graph_Ocases,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,A12: set(A),A23: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A12),A23)
     => ( ( ( A12 = bot_bot(set(A)) )
         => ( A23 != Z ) )
       => ~ ! [X3: A,A7: set(A)] :
              ( ( A12 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),A7) )
             => ! [Y: B] :
                  ( ( A23 = aa(B,B,aa(A,fun(B,B),F2,X3),Y) )
                 => ( ~ member(A,X3,A7)
                   => ~ aa(B,$o,finite_fold_graph(A,B,F2,Z,A7),Y) ) ) ) ) ) ).

% fold_graph.cases
tff(fact_6620_finite__imp__fold__graph,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ? [X_1: B] : aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),X_1) ) ).

% finite_imp_fold_graph
tff(fact_6621_finite__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,finite_finite2(product_prod(A,A)),R2)
     => aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).

% finite_Field
tff(fact_6622_empty__fold__graphE,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B,Xa: B] :
      ( aa(B,$o,finite_fold_graph(A,B,F2,Z,bot_bot(set(A))),Xa)
     => ( Xa = Z ) ) ).

% empty_fold_graphE
tff(fact_6623_fold__graph_OemptyI,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B] : aa(B,$o,finite_fold_graph(A,B,F2,Z,bot_bot(set(A))),Z) ).

% fold_graph.emptyI
tff(fact_6624_FieldI1,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,I),J),R)
     => member(A,I,aa(set(product_prod(A,A)),set(A),field2(A),R)) ) ).

% FieldI1
tff(fact_6625_FieldI2,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,I),J),R)
     => member(A,J,aa(set(product_prod(A,A)),set(A),field2(A),R)) ) ).

% FieldI2
tff(fact_6626_fold__graph_OinsertI,axiom,
    ! [A: $tType,B: $tType,Xa: A,A4: set(A),F2: fun(A,fun(B,B)),Z: B,Ya: B] :
      ( ~ member(A,Xa,A4)
     => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),Ya)
       => aa(B,$o,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),aa(B,B,aa(A,fun(B,B),F2,Xa),Ya)) ) ) ).

% fold_graph.insertI
tff(fact_6627_fold__graph__closed__eq,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B),F2: fun(A,fun(B,B)),G: fun(A,fun(B,B)),Z: B] :
      ( ! [A5: A,B5: B] :
          ( member(A,A5,A4)
         => ( member(B,B5,B2)
           => ( aa(B,B,aa(A,fun(B,B),F2,A5),B5) = aa(B,B,aa(A,fun(B,B),G,A5),B5) ) ) )
     => ( ! [A5: A,B5: B] :
            ( member(A,A5,A4)
           => ( member(B,B5,B2)
             => member(B,aa(B,B,aa(A,fun(B,B),G,A5),B5),B2) ) )
       => ( member(B,Z,B2)
         => ( finite_fold_graph(A,B,F2,Z,A4) = finite_fold_graph(A,B,G,Z,A4) ) ) ) ) ).

% fold_graph_closed_eq
tff(fact_6628_fold__graph__closed__lemma,axiom,
    ! [A: $tType,B: $tType,G: fun(A,fun(B,B)),Z: B,A4: set(A),Xa: B,B2: set(B),F2: fun(A,fun(B,B))] :
      ( aa(B,$o,finite_fold_graph(A,B,G,Z,A4),Xa)
     => ( ! [A5: A,B5: B] :
            ( member(A,A5,A4)
           => ( member(B,B5,B2)
             => ( aa(B,B,aa(A,fun(B,B),F2,A5),B5) = aa(B,B,aa(A,fun(B,B),G,A5),B5) ) ) )
       => ( ! [A5: A,B5: B] :
              ( member(A,A5,A4)
             => ( member(B,B5,B2)
               => member(B,aa(B,B,aa(A,fun(B,B),G,A5),B5),B2) ) )
         => ( member(B,Z,B2)
           => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),Xa)
              & member(B,Xa,B2) ) ) ) ) ) ).

% fold_graph_closed_lemma
tff(fact_6629_mono__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),R2)),aa(set(product_prod(A,A)),set(A),field2(A),S3)) ) ).

% mono_Field
tff(fact_6630_comp__fun__commute__on_Ofold__graph__finite,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Z: B,A4: set(A),Ya: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),Ya)
       => aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% comp_fun_commute_on.fold_graph_finite
tff(fact_6631_comp__fun__commute__on_Ofold__graph__determ,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),Z: B,Xa: B,Ya: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),Xa)
         => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),Ya)
           => ( Ya = Xa ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_determ
tff(fact_6632_fold__graph__image,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(A,B),A4: set(A),F2: fun(B,fun(C,C)),Z: C] :
      ( inj_on(A,B,G,A4)
     => ( finite_fold_graph(B,C,F2,Z,aa(set(A),set(B),image2(A,B,G),A4)) = finite_fold_graph(A,C,aa(fun(A,B),fun(A,fun(C,C)),comp(B,fun(C,C),A,F2),G),Z,A4) ) ) ).

% fold_graph_image
tff(fact_6633_comp__fun__commute__on_Ofold__graph__insertE,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B,V2: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),V2)
         => ( ~ member(A,Xa,A4)
           => ~ ! [Y: B] :
                  ( ( V2 = aa(B,B,aa(A,fun(B,B),F2,Xa),Y) )
                 => ~ aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),Y) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE
tff(fact_6634_comp__fun__commute__on_Ofold__equality,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),Z: B,Ya: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),Ya)
         => ( finite_fold(A,B,F2,Z,A4) = Ya ) ) ) ) ).

% comp_fun_commute_on.fold_equality
tff(fact_6635_Finite__Set_Ofold__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,fun(A,A)),Z: A,A4: set(B)] :
      finite_fold(B,A,F2,Z,A4) = $ite(aa(set(B),$o,finite_finite2(B),A4),the(A,finite_fold_graph(B,A,F2,Z,A4)),Z) ).

% Finite_Set.fold_def
tff(fact_6636_comp__fun__commute__on_Ofold__graph__fold,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => aa(B,$o,finite_fold_graph(A,B,F2,Z,A4),finite_fold(A,B,F2,Z,A4)) ) ) ) ).

% comp_fun_commute_on.fold_graph_fold
tff(fact_6637_Field__natLeq__on,axiom,
    ! [Na: nat] : aa(set(product_prod(nat,nat)),set(nat),field2(nat),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Na)))) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Na)) ).

% Field_natLeq_on
tff(fact_6638_subset__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))))
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),A3),R2) ) ) ) ) ).

% subset_Image1_Image1_iff
tff(fact_6639_preorder__on__empty,axiom,
    ! [A: $tType] : order_preorder_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% preorder_on_empty
tff(fact_6640_subset__Image__Image__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( order_preorder_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),A4)),aa(set(A),set(A),image(A,A,R2),B2))
          <=> ! [X: A] :
                ( member(A,X,A4)
               => ? [Xa3: A] :
                    ( member(A,Xa3,B2)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X),R2) ) ) ) ) ) ) ).

% subset_Image_Image_iff
tff(fact_6641_relChain__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [R2: set(product_prod(A,A)),As: fun(A,B)] :
          ( bNF_Ca3754400796208372196lChain(A,B,R2,As)
        <=> ! [I4: A,J3: A] :
              ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,I4),J3),R2)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,As,I4)),aa(A,B,As,J3)) ) ) ) ).

% relChain_def
tff(fact_6642_natLess__def,axiom,
    bNF_Ca8459412986667044542atLess = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat))) ).

% natLess_def
tff(fact_6643_linear__order__on__singleton,axiom,
    ! [A: $tType,Xa: A] : order_679001287576687338der_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Xa)),bot_bot(set(product_prod(A,A))))) ).

% linear_order_on_singleton
tff(fact_6644_Total__subset__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
       => ( ( R2 = bot_bot(set(product_prod(A,A))) )
          | ? [A5: A] : R2 = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A5),A5)),bot_bot(set(product_prod(A,A)))) ) ) ) ).

% Total_subset_Id
tff(fact_6645_pair__in__Id__conv,axiom,
    ! [A: $tType,A3: A,B3: A] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),id2(A))
    <=> ( A3 = B3 ) ) ).

% pair_in_Id_conv
tff(fact_6646_IdI,axiom,
    ! [A: $tType,A3: A] : member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),A3),id2(A)) ).

% IdI
tff(fact_6647_Image__Id,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),image(A,A,id2(A)),A4) = A4 ).

% Image_Id
tff(fact_6648_total__on__diff__Id,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A4,aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),R2),id2(A)))
    <=> total_on(A,A4,R2) ) ).

% total_on_diff_Id
tff(fact_6649_IdE,axiom,
    ! [A: $tType,P3: product_prod(A,A)] :
      ( member(product_prod(A,A),P3,id2(A))
     => ~ ! [X3: A] : P3 != aa(A,product_prod(A,A),product_Pair(A,A,X3),X3) ) ).

% IdE
tff(fact_6650_relpow_Osimps_I1_J,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(nat,fun(set(product_prod(A,A)),set(product_prod(A,A))),compow(set(product_prod(A,A))),zero_zero(nat)),R) = id2(A) ).

% relpow.simps(1)
tff(fact_6651_Id__def,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_aeb(product_prod(A,A),$o)) ).

% Id_def
tff(fact_6652_lnear__order__on__empty,axiom,
    ! [A: $tType] : order_679001287576687338der_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% lnear_order_on_empty
tff(fact_6653_Total__Id__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( total_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ~ aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),id2(A))
       => ( aa(set(product_prod(A,A)),set(A),field2(A),R2) = aa(set(product_prod(A,A)),set(A),field2(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),R2),id2(A))) ) ) ) ).

% Total_Id_Field
tff(fact_6654_bsqr__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : bNF_Wellorder_bsqr(A,R2) = aa(fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),set(product_prod(product_prod(A,A),product_prod(A,A))),collect(product_prod(product_prod(A,A),product_prod(A,A))),aa(fun(product_prod(A,A),fun(product_prod(A,A),$o)),fun(product_prod(product_prod(A,A),product_prod(A,A)),$o),product_case_prod(product_prod(A,A),product_prod(A,A),$o),aa(fun(A,fun(A,fun(product_prod(A,A),$o))),fun(product_prod(A,A),fun(product_prod(A,A),$o)),product_case_prod(A,A,fun(product_prod(A,A),$o)),aTP_Lamp_aed(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),R2)))) ).

% bsqr_def
tff(fact_6655_max__ext_Omax__extI,axiom,
    ! [A: $tType,X4: set(A),Y3: set(A),R: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),X4)
     => ( aa(set(A),$o,finite_finite2(A),Y3)
       => ( ( Y3 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => ? [Xa2: A] :
                    ( member(A,Xa2,Y3)
                    & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),Xa2),R) ) )
           => member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),X4),Y3),max_ext(A,R)) ) ) ) ) ).

% max_ext.max_extI
tff(fact_6656_max__ext__additive,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),R: set(product_prod(A,A)),C2: set(A),D3: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A4),B2),max_ext(A,R))
     => ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),C2),D3),max_ext(A,R))
       => member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),C2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B2),D3)),max_ext(A,R)) ) ) ).

% max_ext_additive
tff(fact_6657_max__ext_Ocases,axiom,
    ! [A: $tType,A12: set(A),A23: set(A),R: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A12),A23),max_ext(A,R))
     => ~ ( aa(set(A),$o,finite_finite2(A),A12)
         => ( aa(set(A),$o,finite_finite2(A),A23)
           => ( ( A23 != bot_bot(set(A)) )
             => ~ ! [X2: A] :
                    ( member(A,X2,A12)
                   => ? [Xa4: A] :
                        ( member(A,Xa4,A23)
                        & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X2),Xa4),R) ) ) ) ) ) ) ).

% max_ext.cases
tff(fact_6658_max__ext_Osimps,axiom,
    ! [A: $tType,A12: set(A),A23: set(A),R: set(product_prod(A,A))] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A12),A23),max_ext(A,R))
    <=> ( aa(set(A),$o,finite_finite2(A),A12)
        & aa(set(A),$o,finite_finite2(A),A23)
        & ( A23 != bot_bot(set(A)) )
        & ! [X: A] :
            ( member(A,X,A12)
           => ? [Xa3: A] :
                ( member(A,Xa3,A23)
                & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Xa3),R) ) ) ) ) ).

% max_ext.simps
tff(fact_6659_Linear__order__wf__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),R2),id2(A)))
      <=> ! [A8: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),aa(set(product_prod(A,A)),set(A),field2(A),R2))
           => ( ( A8 != bot_bot(set(A)) )
             => ? [X: A] :
                  ( member(A,X,A8)
                  & ! [Xa3: A] :
                      ( member(A,Xa3,A8)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Xa3),R2) ) ) ) ) ) ) ).

% Linear_order_wf_diff_Id
tff(fact_6660_max__ext__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] : max_ext(A,R) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_aee(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R))) ).

% max_ext_eq
tff(fact_6661_bex__empty,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ~ ? [X2: A] :
          ( member(A,X2,bot_bot(set(A)))
          & aa(A,$o,P,X2) ) ).

% bex_empty
tff(fact_6662_finite__Collect__bex,axiom,
    ! [B: $tType,A: $tType,A4: set(A),Q: fun(B,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aef(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),Q)))
      <=> ! [X: A] :
            ( member(A,X,A4)
           => aa(set(B),$o,finite_finite2(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),X))) ) ) ) ).

% finite_Collect_bex
tff(fact_6663_bex__UNIV,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,top_top(set(A)))
          & aa(A,$o,P,X) )
    <=> ? [X_13: A] : aa(A,$o,P,X_13) ) ).

% bex_UNIV
tff(fact_6664_Image__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(B,fun(A,$o)),A4: set(B)] : aa(set(B),set(A),image(B,A,aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P))),A4) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_aeg(fun(B,fun(A,$o)),fun(set(B),fun(A,$o)),P),A4)) ).

% Image_Collect_case_prod
tff(fact_6665_wf__if__measure,axiom,
    ! [A: $tType,P: fun(A,$o),F2: fun(A,nat),G: fun(A,A)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,aa(A,A,G,X3))),aa(A,nat,F2,X3)) )
     => wf(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_aeh(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),P),G)))) ) ).

% wf_if_measure
tff(fact_6666_wf,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => wf(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),ord_less(A)))) ) ).

% wf
tff(fact_6667_wf__less,axiom,
    wf(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less(nat)))) ).

% wf_less
tff(fact_6668_wf__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P3: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),P3),R2)
       => wf(A,P3) ) ) ).

% wf_subset
tff(fact_6669_wfE__min_H,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Q: set(A)] :
      ( wf(A,R)
     => ( ( Q != bot_bot(set(A)) )
       => ~ ! [Z2: A] :
              ( member(A,Z2,Q)
             => ~ ! [Y2: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y2),Z2),R)
                   => ~ member(A,Y2,Q) ) ) ) ) ).

% wfE_min'
tff(fact_6670_Image__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S3: set(B)] : aa(set(B),set(A),image(B,A,R2),S3) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_aei(set(product_prod(B,A)),fun(set(B),fun(A,$o)),R2),S3)) ).

% Image_def
tff(fact_6671_image__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] : aa(set(B),set(A),image2(B,A,F2),A4) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_aej(fun(B,A),fun(set(B),fun(A,$o)),F2),A4)) ).

% image_def
tff(fact_6672_wf__bounded__measure,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,nat),F2: fun(A,nat)] :
      ( ! [A5: A,B5: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B5),A5),R2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Ub,B5)),aa(A,nat,Ub,A5))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,B5)),aa(A,nat,Ub,A5))
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,A5)),aa(A,nat,F2,B5)) ) )
     => wf(A,R2) ) ).

% wf_bounded_measure
tff(fact_6673_wfE__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),A4: set(A)] :
      ( wf(A,R)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),image(A,A,R),A4))
       => ( A4 = bot_bot(set(A)) ) ) ) ).

% wfE_pf
tff(fact_6674_wfI__pf,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A7: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A7),aa(set(A),set(A),image(A,A,R),A7))
         => ( A7 = bot_bot(set(A)) ) )
     => wf(A,R) ) ).

% wfI_pf
tff(fact_6675_wf__linord__ex__has__least,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),P: fun(B,$o),K: B,M: fun(B,A)] :
      ( wf(A,R2)
     => ( ! [X3: A,Y: A] :
            ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y),transitive_trancl(A,R2))
          <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),X3),transitive_rtrancl(A,R2)) )
       => ( aa(B,$o,P,K)
         => ? [X3: B] :
              ( aa(B,$o,P,X3)
              & ! [Y2: B] :
                  ( aa(B,$o,P,Y2)
                 => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,aa(B,A,M,X3)),aa(B,A,M,Y2)),transitive_rtrancl(A,R2)) ) ) ) ) ) ).

% wf_linord_ex_has_least
tff(fact_6676_Bex__fold,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ? [X: A] :
            ( member(A,X,A4)
            & aa(A,$o,P,X) )
      <=> finite_fold(A,$o,aTP_Lamp_aek(fun(A,$o),fun(A,fun($o,$o)),P),$false,A4) ) ) ).

% Bex_fold
tff(fact_6677_nths__nths,axiom,
    ! [A: $tType,Xs: list(A),A4: set(nat),B2: set(nat)] : nths(A,nths(A,Xs,A4),B2) = nths(A,Xs,aa(fun(nat,$o),set(nat),collect(nat),aa(set(nat),fun(nat,$o),aTP_Lamp_aem(set(nat),fun(set(nat),fun(nat,$o)),A4),B2))) ).

% nths_nths
tff(fact_6678_wf__eq__minimal2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( wf(A,R2)
    <=> ! [A8: set(A)] :
          ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),aa(set(product_prod(A,A)),set(A),field2(A),R2))
            & ( A8 != bot_bot(set(A)) ) )
         => ? [X: A] :
              ( member(A,X,A8)
              & ! [Xa3: A] :
                  ( member(A,Xa3,A8)
                 => ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X),R2) ) ) ) ) ).

% wf_eq_minimal2
tff(fact_6679_wf__bounded__set,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),Ub: fun(A,set(B)),F2: fun(A,set(B))] :
      ( ! [A5: A,B5: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B5),A5),R2)
         => ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Ub,A5))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Ub,B5)),aa(A,set(B),Ub,A5))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),F2,B5)),aa(A,set(B),Ub,A5))
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(A,set(B),F2,A5)),aa(A,set(B),F2,B5)) ) )
     => wf(A,R2) ) ).

% wf_bounded_set
tff(fact_6680_finite__subset__wf,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => wf(set(A),aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_aen(set(A),fun(set(A),fun(set(A),$o)),A4)))) ) ).

% finite_subset_wf
tff(fact_6681_min__ext__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : min_ext(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aTP_Lamp_aeo(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),R2)) ).

% min_ext_def
tff(fact_6682_max__extp_Omax__extI,axiom,
    ! [A: $tType,X4: set(A),Y3: set(A),R: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,finite_finite2(A),X4)
     => ( aa(set(A),$o,finite_finite2(A),Y3)
       => ( ( Y3 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => ? [Xa2: A] :
                    ( member(A,Xa2,Y3)
                    & aa(A,$o,aa(A,fun(A,$o),R,X3),Xa2) ) )
           => max_extp(A,R,X4,Y3) ) ) ) ) ).

% max_extp.max_extI
tff(fact_6683_max__extp_Ocases,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),A12: set(A),A23: set(A)] :
      ( max_extp(A,R,A12,A23)
     => ~ ( aa(set(A),$o,finite_finite2(A),A12)
         => ( aa(set(A),$o,finite_finite2(A),A23)
           => ( ( A23 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
             => ~ ! [X2: A] :
                    ( member(A,X2,A12)
                   => ? [Xa4: A] :
                        ( member(A,Xa4,A23)
                        & aa(A,$o,aa(A,fun(A,$o),R,X2),Xa4) ) ) ) ) ) ) ).

% max_extp.cases
tff(fact_6684_max__extp_Osimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),A12: set(A),A23: set(A)] :
      ( max_extp(A,R,A12,A23)
    <=> ( aa(set(A),$o,finite_finite2(A),A12)
        & aa(set(A),$o,finite_finite2(A),A23)
        & ( A23 != aa(fun(A,$o),set(A),collect(A),bot_bot(fun(A,$o))) )
        & ! [X: A] :
            ( member(A,X,A12)
           => ? [Xa3: A] :
                ( member(A,Xa3,A23)
                & aa(A,$o,aa(A,fun(A,$o),R,X),Xa3) ) ) ) ) ).

% max_extp.simps
tff(fact_6685_cauchy__def,axiom,
    ! [X4: fun(nat,rat)] :
      ( cauchy(X4)
    <=> ! [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
         => ? [K3: nat] :
            ! [M2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),M2)
             => ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,minus_minus(rat,aa(nat,rat,X4,M2)),aa(nat,rat,X4,N)))),R5) ) ) ) ) ).

% cauchy_def
tff(fact_6686_cauchyI,axiom,
    ! [X4: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R3)
         => ? [K9: nat] :
            ! [M4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K9),M4)
             => ! [N2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K9),N2)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,minus_minus(rat,aa(nat,rat,X4,M4)),aa(nat,rat,X4,N2)))),R3) ) ) )
     => cauchy(X4) ) ).

% cauchyI
tff(fact_6687_cauchy__imp__bounded,axiom,
    ! [X4: fun(nat,rat)] :
      ( cauchy(X4)
     => ? [B5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B5)
          & ! [N3: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X4,N3))),B5) ) ) ).

% cauchy_imp_bounded
tff(fact_6688_cauchyD,axiom,
    ! [X4: fun(nat,rat),R2: rat] :
      ( cauchy(X4)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
       => ? [K2: nat] :
          ! [M3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),M3)
           => ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(rat,rat,minus_minus(rat,aa(nat,rat,X4,M3)),aa(nat,rat,X4,N3)))),R2) ) ) ) ) ).

% cauchyD
tff(fact_6689_le__Real,axiom,
    ! [X4: fun(nat,rat),Y3: fun(nat,rat)] :
      ( cauchy(X4)
     => ( cauchy(Y3)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real2(X4)),real2(Y3))
        <=> ! [R5: rat] :
              ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
             => ? [K3: nat] :
                ! [N: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
                 => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X4,N)),aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),aa(nat,rat,Y3,N)),R5)) ) ) ) ) ) ).

% le_Real
tff(fact_6690_cauchy__not__vanishes,axiom,
    ! [X4: fun(nat,rat)] :
      ( cauchy(X4)
     => ( ~ vanishes(X4)
       => ? [B5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B5)
            & ? [K2: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B5),aa(rat,rat,abs_abs(rat),aa(nat,rat,X4,N3))) ) ) ) ) ).

% cauchy_not_vanishes
tff(fact_6691_vanishes__mult__bounded,axiom,
    ! [X4: fun(nat,rat),Y3: fun(nat,rat)] :
      ( ? [A10: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),A10)
          & ! [N2: nat] : aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X4,N2))),A10) )
     => ( vanishes(Y3)
       => vanishes(aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aep(fun(nat,rat),fun(fun(nat,rat),fun(nat,rat)),X4),Y3)) ) ) ).

% vanishes_mult_bounded
tff(fact_6692_vanishesD,axiom,
    ! [X4: fun(nat,rat),R2: rat] :
      ( vanishes(X4)
     => ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2)
       => ? [K2: nat] :
          ! [N3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
           => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X4,N3))),R2) ) ) ) ).

% vanishesD
tff(fact_6693_vanishesI,axiom,
    ! [X4: fun(nat,rat)] :
      ( ! [R3: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R3)
         => ? [K9: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K9),N2)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X4,N2))),R3) ) )
     => vanishes(X4) ) ).

% vanishesI
tff(fact_6694_vanishes__def,axiom,
    ! [X4: fun(nat,rat)] :
      ( vanishes(X4)
    <=> ! [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
         => ? [K3: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),aa(rat,rat,abs_abs(rat),aa(nat,rat,X4,N))),R5) ) ) ) ).

% vanishes_def
tff(fact_6695_cauchy__not__vanishes__cases,axiom,
    ! [X4: fun(nat,rat)] :
      ( cauchy(X4)
     => ( ~ vanishes(X4)
       => ? [B5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),B5)
            & ? [K2: nat] :
                ( ! [N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B5),aa(rat,rat,uminus_uminus(rat),aa(nat,rat,X4,N3))) )
                | ! [N3: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),N3)
                   => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),B5),aa(nat,rat,X4,N3)) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
tff(fact_6696_not__positive__Real,axiom,
    ! [X4: fun(nat,rat)] :
      ( cauchy(X4)
     => ( ~ aa(real,$o,positive2,real2(X4))
      <=> ! [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
           => ? [K3: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(nat,rat,X4,N)),R5) ) ) ) ) ).

% not_positive_Real
tff(fact_6697_positive__Real,axiom,
    ! [X4: fun(nat,rat)] :
      ( cauchy(X4)
     => ( aa(real,$o,positive2,real2(X4))
      <=> ? [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
            & ? [K3: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,X4,N)) ) ) ) ) ).

% positive_Real
tff(fact_6698_less__real__def,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
    <=> aa(real,$o,positive2,aa(real,real,minus_minus(real,Ya),Xa)) ) ).

% less_real_def
tff(fact_6699_Real_Opositive_Orep__eq,axiom,
    ! [Xa: real] :
      ( aa(real,$o,positive2,Xa)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K3: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,aa(real,fun(nat,rat),rep_real,Xa),N)) ) ) ) ).

% Real.positive.rep_eq
tff(fact_6700_finite__def,axiom,
    ! [A: $tType] : finite_finite2(A) = complete_lattice_lfp(fun(set(A),$o),aTP_Lamp_zr(fun(set(A),$o),fun(set(A),$o))) ).

% finite_def
tff(fact_6701_lfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),Xa: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F3)
         => ( ( aa(A,A,F3,Xa) = Xa )
           => ( ! [Z2: A] :
                  ( ( aa(A,A,F3,Z2) = Z2 )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z2) )
             => ( complete_lattice_lfp(A,F3) = Xa ) ) ) ) ) ).

% lfp_eqI
tff(fact_6702_lfp__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,fun(A,A))] :
          ( ! [X3: A,Y: A,W: A,Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),W)),aa(A,A,aa(A,fun(A,A),F2,Y),Z2)) ) )
         => ( complete_lattice_lfp(A,aTP_Lamp_aeq(fun(A,fun(A,A)),fun(A,A),F2)) = complete_lattice_lfp(A,aTP_Lamp_aer(fun(A,fun(A,A)),fun(A,A),F2)) ) ) ) ).

% lfp_lfp
tff(fact_6703_lfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),G: fun(A,A)] :
          ( ! [Z8: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,Z8)),aa(A,A,G,Z8))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),complete_lattice_lfp(A,G)) ) ) ).

% lfp_mono
tff(fact_6704_lfp__lowerbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),A4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,A4)),A4)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),A4) ) ) ).

% lfp_lowerbound
tff(fact_6705_lfp__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),A4: A] :
          ( ! [U3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,U3)),U3)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),U3) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),complete_lattice_lfp(A,F2)) ) ) ).

% lfp_greatest
tff(fact_6706_lfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A)] : complete_lattice_lfp(A,F2) = aa(set(A),A,complete_Inf_Inf(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aes(fun(A,A),fun(A,$o),F2))) ) ).

% lfp_def
tff(fact_6707_def__lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: A,F2: fun(A,A),P: A] :
          ( ( A4 = complete_lattice_lfp(A,F2) )
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),A4),P))),P)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),P) ) ) ) ) ).

% def_lfp_induct
tff(fact_6708_lfp__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),inf_inf(A),complete_lattice_lfp(A,F2)),P))),P)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),P) ) ) ) ).

% lfp_induct
tff(fact_6709_lfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: fun(A,$o)] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( ! [S2: A] :
                ( aa(A,$o,P,S2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S2),complete_lattice_lfp(A,F2))
                 => aa(A,$o,P,aa(A,A,F2,S2)) ) )
           => ( ! [M9: set(A)] :
                  ( ! [X2: A] :
                      ( member(A,X2,M9)
                     => aa(A,$o,P,X2) )
                 => aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),M9)) )
             => aa(A,$o,P,complete_lattice_lfp(A,F2)) ) ) ) ) ).

% lfp_ordinal_induct
tff(fact_6710_lfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Na: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( complete_lattice_lfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Na)),F2)) = complete_lattice_lfp(A,F2) ) ) ) ).

% lfp_funpow
tff(fact_6711_lfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F2),bot_bot(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A)) )
           => ( complete_lattice_lfp(A,F2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),bot_bot(A)) ) ) ) ) ).

% lfp_Kleene_iter
tff(fact_6712_iteratesp__def,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X2: fun(A,A)] : comple7512665784863727008ratesp(A,X2) = complete_lattice_lfp(fun(A,$o),aTP_Lamp_aat(fun(A,A),fun(fun(A,$o),fun(A,$o)),X2)) ) ).

% iteratesp_def
tff(fact_6713_lfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P: fun(A,$o),F2: fun(A,A),Alpha: fun(A,B),G: fun(B,B)] :
          ( aa(A,$o,P,bot_bot(A))
         => ( ! [X3: A] :
                ( aa(A,$o,P,X3)
               => aa(A,$o,P,aa(A,A,F2,X3)) )
           => ( ! [M9: fun(nat,A)] :
                  ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M9,I3))
                 => aa(A,$o,P,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,M9),top_top(set(nat))))) )
             => ( ! [M9: fun(nat,A)] :
                    ( aa(fun(nat,A),$o,order_mono(nat,A),M9)
                   => ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M9,I3))
                     => ( aa(A,B,Alpha,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,M9),top_top(set(nat))))) = aa(set(B),B,complete_Sup_Sup(B),aa(set(nat),set(B),image2(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aet(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M9)),top_top(set(nat)))) ) ) )
               => ( order_sup_continuous(A,A,F2)
                 => ( order_sup_continuous(B,B,G)
                   => ( ! [X3: A] :
                          ( aa(A,$o,P,X3)
                         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),complete_lattice_lfp(A,F2))
                           => ( aa(A,B,Alpha,aa(A,A,F2,X3)) = aa(B,B,G,aa(A,B,Alpha,X3)) ) ) )
                     => ( ! [X3: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G,X3))
                       => ( aa(A,B,Alpha,complete_lattice_lfp(A,F2)) = complete_lattice_lfp(B,G) ) ) ) ) ) ) ) ) ) ) ).

% lfp_transfer_bounded
tff(fact_6714_sup__continuous__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F2: fun(A,B),G: fun(A,B)] :
          ( order_sup_continuous(A,B,F2)
         => ( order_sup_continuous(A,B,G)
           => order_sup_continuous(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeu(fun(A,B),fun(fun(A,B),fun(A,B)),F2),G)) ) ) ) ).

% sup_continuous_sup
tff(fact_6715_lfp__induct2,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,F2: fun(set(product_prod(A,B)),set(product_prod(A,B))),P: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),complete_lattice_lfp(set(product_prod(A,B)),F2))
     => ( aa(fun(set(product_prod(A,B)),set(product_prod(A,B))),$o,order_mono(set(product_prod(A,B)),set(product_prod(A,B))),F2)
       => ( ! [A5: A,B5: B] :
              ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A5),B5),aa(set(product_prod(A,B)),set(product_prod(A,B)),F2,aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),complete_lattice_lfp(set(product_prod(A,B)),F2)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)))))
             => aa(B,$o,aa(A,fun(B,$o),P,A5),B5) )
         => aa(B,$o,aa(A,fun(B,$o),P,A3),B3) ) ) ) ).

% lfp_induct2
tff(fact_6716_lfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [Alpha: fun(A,B),F2: fun(A,A),G: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( order_sup_continuous(A,A,F2)
           => ( order_sup_continuous(B,B,G)
             => ( ! [X3: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Alpha,bot_bot(A))),aa(B,B,G,X3))
               => ( ! [X3: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),complete_lattice_lfp(A,F2))
                     => ( aa(A,B,Alpha,aa(A,A,F2,X3)) = aa(B,B,G,aa(A,B,Alpha,X3)) ) )
                 => ( aa(A,B,Alpha,complete_lattice_lfp(A,F2)) = complete_lattice_lfp(B,G) ) ) ) ) ) ) ) ).

% lfp_transfer
tff(fact_6717_cclfp__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta3822494911875563373attice(A) )
     => ! [Alpha: fun(A,B),F2: fun(A,A),G: fun(B,B)] :
          ( order_sup_continuous(A,B,Alpha)
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => ( ( aa(A,B,Alpha,bot_bot(A)) = bot_bot(B) )
             => ( ! [X3: A] : aa(A,B,Alpha,aa(A,A,F2,X3)) = aa(B,B,G,aa(A,B,Alpha,X3))
               => ( aa(A,B,Alpha,order_532582986084564980_cclfp(A,F2)) = order_532582986084564980_cclfp(B,G) ) ) ) ) ) ) ).

% cclfp_transfer
tff(fact_6718_iteratesp_Osimps,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),A3: A] :
          ( aa(A,$o,comple7512665784863727008ratesp(A,F2),A3)
        <=> ( ? [X: A] :
                ( ( A3 = aa(A,A,F2,X) )
                & aa(A,$o,comple7512665784863727008ratesp(A,F2),X) )
            | ? [M8: set(A)] :
                ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X: A] :
                    ( member(A,X,M8)
                   => aa(A,$o,comple7512665784863727008ratesp(A,F2),X) ) ) ) ) ) ).

% iteratesp.simps
tff(fact_6719_iteratesp_Ocases,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F2: fun(A,A),A3: A] :
          ( aa(A,$o,comple7512665784863727008ratesp(A,F2),A3)
         => ( ! [X3: A] :
                ( ( A3 = aa(A,A,F2,X3) )
               => ~ aa(A,$o,comple7512665784863727008ratesp(A,F2),X3) )
           => ~ ! [M9: set(A)] :
                  ( ( A3 = aa(set(A),A,complete_Sup_Sup(A),M9) )
                 => ( comple1602240252501008431_chain(A,ord_less_eq(A),M9)
                   => ~ ! [X2: A] :
                          ( member(A,X2,M9)
                         => aa(A,$o,comple7512665784863727008ratesp(A,F2),X2) ) ) ) ) ) ) ).

% iteratesp.cases
tff(fact_6720_iteratesp_OSup,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [M5: set(A),F2: fun(A,A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),M5)
         => ( ! [X3: A] :
                ( member(A,X3,M5)
               => aa(A,$o,comple7512665784863727008ratesp(A,F2),X3) )
           => aa(A,$o,comple7512665784863727008ratesp(A,F2),aa(set(A),A,complete_Sup_Sup(A),M5)) ) ) ) ).

% iteratesp.Sup
tff(fact_6721_sup__continuous__lfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A)] :
          ( order_sup_continuous(A,A,F3)
         => ( complete_lattice_lfp(A,F3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image2(nat,A,aTP_Lamp_aev(fun(A,A),fun(nat,A),F3)),top_top(set(nat)))) ) ) ) ).

% sup_continuous_lfp
tff(fact_6722_ord__class_Olexordp__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( ord_lexordp(A) = complete_lattice_lfp(fun(list(A),fun(list(A),$o)),aTP_Lamp_abu(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)))) ) ) ).

% ord_class.lexordp_def
tff(fact_6723_butlast__take,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( butlast(A,take(A,Na,Xs)) = take(A,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_6724_lexordp__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Xs: list(A),Ya: A,Ys2: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),cons(A,Xa),Xs)),aa(list(A),list(A),cons(A,Ya),Ys2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
              & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2) ) ) ) ) ).

% lexordp_simps(3)
tff(fact_6725_lexordp__append__leftD,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A),Us: list(A),Vs: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),append(A,Xs,Us)),append(A,Xs,Vs))
         => ( ! [A5: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),A5),A5)
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Us),Vs) ) ) ) ).

% lexordp_append_leftD
tff(fact_6726_lexordp__irreflexive,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),X3)
         => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Xs) ) ) ).

% lexordp_irreflexive
tff(fact_6727_lexordp_OCons,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Ya: A,Xs: list(A),Ys2: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),cons(A,Xa),Xs)),aa(list(A),list(A),cons(A,Ya),Ys2)) ) ) ).

% lexordp.Cons
tff(fact_6728_lexordp_OCons__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Ya: A,Xs: list(A),Ys2: list(A)] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Ya),Xa)
           => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
             => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),aa(list(A),list(A),cons(A,Xa),Xs)),aa(list(A),list(A),cons(A,Ya),Ys2)) ) ) ) ) ).

% lexordp.Cons_eq
tff(fact_6729_lexordp__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A),P: fun(list(A),fun(list(A),$o))] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
         => ( ! [Y: A,Ys3: list(A)] : aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),aa(list(A),list(A),cons(A,Y),Ys3))
           => ( ! [X3: A,Xs2: list(A),Y: A,Ys3: list(A)] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                 => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),cons(A,X3),Xs2)),aa(list(A),list(A),cons(A,Y),Ys3)) )
             => ( ! [X3: A,Xs2: list(A),Ys3: list(A)] :
                    ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3)
                   => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs2),Ys3)
                     => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),cons(A,X3),Xs2)),aa(list(A),list(A),cons(A,X3),Ys3)) ) )
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Xs),Ys2) ) ) ) ) ) ).

% lexordp_induct
tff(fact_6730_lexordp__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
         => ( ( ( Xs = nil(A) )
             => ! [Y: A,Ys5: list(A)] : Ys2 != aa(list(A),list(A),cons(A,Y),Ys5) )
           => ( ! [X3: A] :
                  ( ? [Xs4: list(A)] : Xs = aa(list(A),list(A),cons(A,X3),Xs4)
                 => ! [Y: A] :
                      ( ? [Ys5: list(A)] : Ys2 = aa(list(A),list(A),cons(A,Y),Ys5)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y) ) )
             => ~ ! [X3: A,Xs4: list(A)] :
                    ( ( Xs = aa(list(A),list(A),cons(A,X3),Xs4) )
                   => ! [Ys5: list(A)] :
                        ( ( Ys2 = aa(list(A),list(A),cons(A,X3),Ys5) )
                       => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs4),Ys5) ) ) ) ) ) ) ).

% lexordp_cases
tff(fact_6731_lexordp_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A12: list(A),A23: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A12),A23)
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( A12 = nil(A) )
                & ( A23 = aa(list(A),list(A),cons(A,Y4),Ys4) ) )
            | ? [X: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A12 = aa(list(A),list(A),cons(A,X),Xs3) )
                & ( A23 = aa(list(A),list(A),cons(A,Y4),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y4) )
            | ? [X: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( A12 = aa(list(A),list(A),cons(A,X),Xs3) )
                & ( A23 = aa(list(A),list(A),cons(A,Y4),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs3),Ys4) ) ) ) ) ).

% lexordp.simps
tff(fact_6732_lexordp_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A12: list(A),A23: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),A12),A23)
         => ( ( ( A12 = nil(A) )
             => ! [Y: A,Ys3: list(A)] : A23 != aa(list(A),list(A),cons(A,Y),Ys3) )
           => ( ! [X3: A] :
                  ( ? [Xs2: list(A)] : A12 = aa(list(A),list(A),cons(A,X3),Xs2)
                 => ! [Y: A] :
                      ( ? [Ys3: list(A)] : A23 = aa(list(A),list(A),cons(A,Y),Ys3)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y) ) )
             => ~ ! [X3: A,Y: A,Xs2: list(A)] :
                    ( ( A12 = aa(list(A),list(A),cons(A,X3),Xs2) )
                   => ! [Ys3: list(A)] :
                        ( ( A23 = aa(list(A),list(A),cons(A,Y),Ys3) )
                       => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X3),Y)
                         => ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y),X3)
                           => ~ aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs2),Ys3) ) ) ) ) ) ) ) ) ).

% lexordp.cases
tff(fact_6733_lexordp__append__left__rightI,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xa: A,Ya: A,Us: list(A),Xs: list(A),Ys2: list(A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),Xa),Ya)
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),append(A,Us,aa(list(A),list(A),cons(A,Xa),Xs))),append(A,Us,aa(list(A),list(A),cons(A,Ya),Ys2))) ) ) ).

% lexordp_append_left_rightI
tff(fact_6734_lexordp__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
        <=> ( ? [X: A,Vs2: list(A)] : Ys2 = append(A,Xs,aa(list(A),list(A),cons(A,X),Vs2))
            | ? [Us2: list(A),A9: A,B7: A,Vs2: list(A),Ws: list(A)] :
                ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),A9),B7)
                & ( Xs = append(A,Us2,aa(list(A),list(A),cons(A,A9),Vs2)) )
                & ( Ys2 = append(A,Us2,aa(list(A),list(A),cons(A,B7),Ws)) ) ) ) ) ) ).

% lexordp_iff
tff(fact_6735_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),butlast(A,Xs)) ) ) ) ).

% sorted_butlast
tff(fact_6736_nth__butlast,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),butlast(A,Xs)))
     => ( aa(nat,A,nth(A,butlast(A,Xs)),Na) = aa(nat,A,nth(A,Xs),Na) ) ) ).

% nth_butlast
tff(fact_6737_take__butlast,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( take(A,Na,butlast(A,Xs)) = take(A,Na,Xs) ) ) ).

% take_butlast
tff(fact_6738_lexordp__conv__lexord,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys2: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),ord_lexordp(A),Xs),Ys2)
        <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys2),lexord(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),ord_less(A))))) ) ) ).

% lexordp_conv_lexord
tff(fact_6739_finite__refines__card__le,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A4,R))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
       => ( equiv_equiv(A,A4,R)
         => ( equiv_equiv(A,A4,S)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A4,S))),aa(set(set(A)),nat,finite_card(set(A)),equiv_quotient(A,A4,R))) ) ) ) ) ).

% finite_refines_card_le
tff(fact_6740_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,Na))
        <=> ( Na = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_6741_mask__nat__positive__iff,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ).

% mask_nat_positive_iff
tff(fact_6742_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_6743_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Na: nat] :
          ( ( bit_se2239418461657761734s_mask(A,Na) = zero_zero(A) )
        <=> ( Na = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_6744_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_6745_in__quotient__imp__subset,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X4: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( member(set(A),X4,equiv_quotient(A,A4,R2))
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),A4) ) ) ).

% in_quotient_imp_subset
tff(fact_6746_in__quotient__imp__non__empty,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X4: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( member(set(A),X4,equiv_quotient(A,A4,R2))
       => ( X4 != bot_bot(set(A)) ) ) ) ).

% in_quotient_imp_non_empty
tff(fact_6747_less__eq__mask,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),bit_se2239418461657761734s_mask(nat,Na)) ).

% less_eq_mask
tff(fact_6748_mask__nonnegative__int,axiom,
    ! [Na: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,Na)) ).

% mask_nonnegative_int
tff(fact_6749_not__mask__negative__int,axiom,
    ! [Na: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_se2239418461657761734s_mask(int,Na)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_6750_equiv__class__self,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( member(A,A3,A4)
       => member(A,A3,aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) ) ) ).

% equiv_class_self
tff(fact_6751_quotient__disj,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X4: set(A),Y3: set(A)] :
      ( equiv_equiv(A,A4,R2)
     => ( member(set(A),X4,equiv_quotient(A,A4,R2))
       => ( member(set(A),Y3,equiv_quotient(A,A4,R2))
         => ( ( X4 = Y3 )
            | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),X4),Y3) = bot_bot(set(A)) ) ) ) ) ) ).

% quotient_disj
tff(fact_6752_less__mask,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),Na)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),bit_se2239418461657761734s_mask(nat,Na)) ) ).

% less_mask
tff(fact_6753_finite__refines__finite,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A4,R))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
       => ( equiv_equiv(A,A4,R)
         => ( equiv_equiv(A,A4,S)
           => aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A4,S)) ) ) ) ) ).

% finite_refines_finite
tff(fact_6754_equiv__class__eq__iff,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),Xa: A,Ya: A] :
      ( equiv_equiv(A,A4,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2)
      <=> ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A)))) )
          & member(A,Xa,A4)
          & member(A,Ya,A4) ) ) ) ).

% equiv_class_eq_iff
tff(fact_6755_eq__equiv__class__iff,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),Xa: A,Ya: A] :
      ( equiv_equiv(A,A4,R2)
     => ( member(A,Xa,A4)
       => ( member(A,Ya,A4)
         => ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A)))) )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2) ) ) ) ) ).

% eq_equiv_class_iff
tff(fact_6756_equiv__class__eq,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2)
       => ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) ) ) ) ).

% equiv_class_eq
tff(fact_6757_eq__equiv__class,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A,A4: set(A)] :
      ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) )
     => ( equiv_equiv(A,A4,R2)
       => ( member(A,B3,A4)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2) ) ) ) ).

% eq_equiv_class
tff(fact_6758_eq__equiv__class__iff2,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),Xa: A,Ya: A] :
      ( equiv_equiv(A,A4,R2)
     => ( member(A,Xa,A4)
       => ( member(A,Ya,A4)
         => ( ( equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))),R2) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A))),R2) )
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2) ) ) ) ) ).

% eq_equiv_class_iff2
tff(fact_6759_refines__equiv__class__eq2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),A4: set(A),A3: A] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
     => ( equiv_equiv(A,A4,R)
       => ( equiv_equiv(A,A4,S)
         => ( aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq2
tff(fact_6760_refines__equiv__class__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),A4: set(A),A3: A] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
     => ( equiv_equiv(A,A4,R)
       => ( equiv_equiv(A,A4,S)
         => ( aa(set(A),set(A),image(A,A,R),aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))) = aa(set(A),set(A),image(A,A,S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ) ).

% refines_equiv_class_eq
tff(fact_6761_equiv__imp__dvd__card,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),K: nat] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( equiv_equiv(A,A4,R2)
       => ( ! [X6: set(A)] :
              ( member(set(A),X6,equiv_quotient(A,A4,R2))
             => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),X6)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),K),aa(set(A),nat,finite_card(A),A4)) ) ) ) ).

% equiv_imp_dvd_card
tff(fact_6762_refines__equiv__image__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),S)
     => ( equiv_equiv(A,A4,R)
       => ( equiv_equiv(A,A4,S)
         => ( aa(set(set(A)),set(set(A)),image2(set(A),set(A),image(A,A,S)),equiv_quotient(A,A4,R)) = equiv_quotient(A,A4,S) ) ) ) ) ).

% refines_equiv_image_eq
tff(fact_6763_subset__equiv__class,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),B3: A,A3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))
       => ( member(A,B3,A4)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2) ) ) ) ).

% subset_equiv_class
tff(fact_6764_equiv__class__subset,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) ) ) ).

% equiv_class_subset
tff(fact_6765_equiv__class__nondisjoint,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),Xa: A,A3: A,B3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( member(A,Xa,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,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))))
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2) ) ) ).

% equiv_class_nondisjoint
tff(fact_6766_in__quotient__imp__in__rel,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X4: set(A),Xa: A,Ya: A] :
      ( equiv_equiv(A,A4,R2)
     => ( member(set(A),X4,equiv_quotient(A,A4,R2))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A))))),X4)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2) ) ) ) ).

% in_quotient_imp_in_rel
tff(fact_6767_mask__nat__less__exp,axiom,
    ! [Na: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),bit_se2239418461657761734s_mask(nat,Na)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na)) ).

% mask_nat_less_exp
tff(fact_6768_UN__equiv__class2,axiom,
    ! [A: $tType,C: $tType,B: $tType,A15: set(A),R12: set(product_prod(A,A)),A25: set(B),R23: set(product_prod(B,B)),F2: fun(A,fun(B,set(C))),A12: A,A23: B] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A25,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F2)
         => ( member(A,A12,A15)
           => ( member(B,A23,A25)
             => ( aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image2(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_aew(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F2),A23)),aa(set(A),set(A),image(A,A,R12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A12),bot_bot(set(A)))))) = aa(B,set(C),aa(A,fun(B,set(C)),F2,A12),A23) ) ) ) ) ) ) ).

% UN_equiv_class2
tff(fact_6769_UN__equiv__class,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(A,A)),F2: fun(A,set(B)),A3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( equiv_congruent(A,set(B),R2,F2)
       => ( member(A,A3,A4)
         => ( aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) = aa(A,set(B),F2,A3) ) ) ) ) ).

% UN_equiv_class
tff(fact_6770_congruent2__implies__congruent__UN,axiom,
    ! [A: $tType,C: $tType,B: $tType,A15: set(A),R12: set(product_prod(A,A)),A25: set(B),R23: set(product_prod(B,B)),F2: fun(A,fun(B,set(C))),A3: B] :
      ( equiv_equiv(A,A15,R12)
     => ( equiv_equiv(B,A25,R23)
       => ( equiv_congruent2(A,B,set(C),R12,R23,F2)
         => ( member(B,A3,A25)
           => equiv_congruent(A,set(C),R12,aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_aew(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),R23),F2),A3)) ) ) ) ) ).

% congruent2_implies_congruent_UN
tff(fact_6771_proj__iff,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),Xa: A,Ya: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A))))),A4)
       => ( ( aa(A,set(A),equiv_proj(A,A,R2),Xa) = aa(A,set(A),equiv_proj(A,A,R2),Ya) )
        <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2) ) ) ) ).

% proj_iff
tff(fact_6772_independentD,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A),Ta: set(A),U: fun(A,real),V2: A] :
          ( ~ real_V358717886546972837endent(A,S3)
         => ( aa(set(A),$o,finite_finite2(A),Ta)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S3)
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),U)),Ta) = zero_zero(A) )
               => ( member(A,V2,Ta)
                 => ( aa(A,real,U,V2) = zero_zero(real) ) ) ) ) ) ) ) ).

% independentD
tff(fact_6773_independent__empty,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ~ real_V358717886546972837endent(A,bot_bot(set(A))) ) ).

% independent_empty
tff(fact_6774_dependent__single,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A] :
          ( real_V358717886546972837endent(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))
        <=> ( Xa = zero_zero(A) ) ) ) ).

% dependent_single
tff(fact_6775_dependent__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A),A4: set(A)] :
          ( real_V358717886546972837endent(A,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
           => real_V358717886546972837endent(A,A4) ) ) ) ).

% dependent_mono
tff(fact_6776_independent__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A),B2: set(A)] :
          ( ~ real_V358717886546972837endent(A,A4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
           => ~ real_V358717886546972837endent(A,B2) ) ) ) ).

% independent_mono
tff(fact_6777_independent__Union__directed,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [C2: set(set(A))] :
          ( ! [C5: set(A),D6: set(A)] :
              ( member(set(A),C5,C2)
             => ( member(set(A),D6,C2)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),D6)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D6),C5) ) ) )
         => ( ! [C5: set(A)] :
                ( member(set(A),C5,C2)
               => ~ real_V358717886546972837endent(A,C5) )
           => ~ real_V358717886546972837endent(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2)) ) ) ) ).

% independent_Union_directed
tff(fact_6778_dependent__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A)] :
          ( member(A,zero_zero(A),A4)
         => real_V358717886546972837endent(A,A4) ) ) ).

% dependent_zero
tff(fact_6779_unique__representation,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),F2: fun(A,real),G: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( ! [V3: A] :
                ( ( aa(A,real,F2,V3) != zero_zero(real) )
               => member(A,V3,Basis) )
           => ( ! [V3: A] :
                  ( ( aa(A,real,G,V3) != zero_zero(real) )
                 => member(A,V3,Basis) )
             => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F2)))
               => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),G)))
                 => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),F2)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F2))) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),G)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),G))) )
                   => ( F2 = G ) ) ) ) ) ) ) ) ).

% unique_representation
tff(fact_6780_dependent__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( real_V358717886546972837endent(A,S)
          <=> ? [U6: fun(A,real)] :
                ( ? [X: A] :
                    ( member(A,X,S)
                    & ( aa(A,real,U6,X) != zero_zero(real) ) )
                & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),U6)),S) = zero_zero(A) ) ) ) ) ) ).

% dependent_finite
tff(fact_6781_independent__if__scalars__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [F5: fun(A,real),X3: A] :
                ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),F5)),A4) = zero_zero(A) )
               => ( member(A,X3,A4)
                 => ( aa(A,real,F5,X3) = zero_zero(real) ) ) )
           => ~ real_V358717886546972837endent(A,A4) ) ) ) ).

% independent_if_scalars_zero
tff(fact_6782_independentD__unique,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A),X4: fun(A,real),Y3: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,B2)
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X4)))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X4))),B2)
             => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),Y3)))
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),Y3))),B2)
                 => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),X4)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X4))) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),Y3)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),Y3))) )
                   => ( X4 = Y3 ) ) ) ) ) ) ) ) ).

% independentD_unique
tff(fact_6783_proj__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),Xa: B] : aa(B,set(A),equiv_proj(B,A,R2),Xa) = aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B)))) ).

% proj_def
tff(fact_6784_independent__explicit__finite__subsets,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A)] :
          ( ~ real_V358717886546972837endent(A,A4)
        <=> ! [S10: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S10),A4)
             => ( aa(set(A),$o,finite_finite2(A),S10)
               => ! [U6: fun(A,real)] :
                    ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),U6)),S10) = zero_zero(A) )
                   => ! [X: A] :
                        ( member(A,X,S10)
                       => ( aa(A,real,U6,X) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_finite_subsets
tff(fact_6785_independent__explicit__module,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( ~ real_V358717886546972837endent(A,S3)
        <=> ! [T2: set(A),U6: fun(A,real),V6: A] :
              ( aa(set(A),$o,finite_finite2(A),T2)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),S3)
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),U6)),T2) = zero_zero(A) )
                 => ( member(A,V6,T2)
                   => ( aa(A,real,U6,V6) = zero_zero(real) ) ) ) ) ) ) ) ).

% independent_explicit_module
tff(fact_6786_dependent__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( real_V358717886546972837endent(A,S3)
        <=> ? [T2: set(A)] :
              ( aa(set(A),$o,finite_finite2(A),T2)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),S3)
              & ? [U6: fun(A,real)] :
                  ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),U6)),T2) = zero_zero(A) )
                  & ? [X: A] :
                      ( member(A,X,T2)
                      & ( aa(A,real,U6,X) != zero_zero(real) ) ) ) ) ) ) ).

% dependent_explicit
tff(fact_6787_independentD__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A),X4: fun(A,real),Xa: A] :
          ( ~ real_V358717886546972837endent(A,B2)
         => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X4)))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X4))),B2)
             => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),X4)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X4))) = zero_zero(A) )
               => ( aa(A,real,X4,Xa) = zero_zero(real) ) ) ) ) ) ) ).

% independentD_alt
tff(fact_6788_independent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] :
          ( ~ real_V358717886546972837endent(A,B2)
        <=> ! [X8: fun(A,real)] :
              ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X8)))
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X8))),B2)
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),X8)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X8))) = zero_zero(A) )
                 => ! [X: A] : aa(A,real,X8,X) = zero_zero(real) ) ) ) ) ) ).

% independent_alt
tff(fact_6789_dependent__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] :
          ( real_V358717886546972837endent(A,B2)
        <=> ? [X8: fun(A,real)] :
              ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X8)))
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X8))),B2)
              & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),X8)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),X8))) = zero_zero(A) )
              & ? [X: A] : aa(A,real,X8,X) != zero_zero(real) ) ) ) ).

% dependent_alt
tff(fact_6790_isUCont__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,top_top(set(A)),F2)
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => ? [S7: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),S7)
                  & ! [X: A,Y4: A] :
                      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,Y4)),S7)
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,X),aa(A,B,F2,Y4))),R5) ) ) ) ) ) ).

% isUCont_def
tff(fact_6791_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Na: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,Na)
        <=> ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_6792_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat)) ) ).

% possible_bit_0
tff(fact_6793_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,I)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_6794_uniformly__continuous__on__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [S3: set(A),F2: fun(A,B)] :
          ( topolo6026614971017936543ous_on(A,B,S3,F2)
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [D5: real] :
                  ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
                  & ! [X: A] :
                      ( member(A,X,S3)
                     => ! [Xa3: A] :
                          ( member(A,Xa3,S3)
                         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Xa3,X)),D5)
                           => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F2,Xa3),aa(A,B,F2,X))),E3) ) ) ) ) ) ) ) ).

% uniformly_continuous_on_def
tff(fact_6795_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] :
          aa(A,A,bit_se4197421643247451524op_bit(A,M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
                & bit_se6407376104438227557le_bit(A,type2(A),Na) ))),
            aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,Na),M))) ) ).

% drop_bit_exp_eq
tff(fact_6796_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na) ) ) ) ).

% bit_minus_2_iff
tff(fact_6797_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_6798_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_6799_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,bit_se2239418461657761734s_mask(A,M)),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% bit_mask_iff
tff(fact_6800_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C3: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C3) = zero_zero(A) )
         => ( ! [X3: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X3) = zero_zero(A) )
               => aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),C3),X3) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C3 ) ) ) ) ).

% CHAR_eqI
tff(fact_6801_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Na) = zero_zero(A) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),semiri4206861660011772517g_char(A,type2(A))),Na) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_6802_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
           => ( aa(nat,A,semiring_1_of_nat(A),N) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_6803_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),C3)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C3) = zero_zero(A) )
           => ( ! [X3: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),X3)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X3),C3)
                   => ( aa(nat,A,semiring_1_of_nat(A),X3) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C3 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_6804_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),N)
            & ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_6805_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4730199178511100633sh_bit(A,M),A3)),Na)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
            & bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ) ).

% bit_push_bit_iff
tff(fact_6806_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Na: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) = zero_zero(A) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ).

% fold_possible_bit
tff(fact_6807_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ) ) ).

% bit_minus_exp_iff
tff(fact_6808_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),Na)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Na)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_6809_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),Na)
        <=> ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),aa(nat,nat,minus_minus(nat,Na),one_one(nat)))
            & ( Na != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),Na) ) ) ) ).

% bit_double_iff
tff(fact_6810_rat__less__eq__code,axiom,
    ! [P3: rat,Q5: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P3),Q5)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_afa(rat,fun(int,fun(int,$o)),Q5)),quotient_of(P3)) ) ).

% rat_less_eq_code
tff(fact_6811_of__real__sqrt,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( real_Vector_of_real(complex,aa(real,real,sqrt,Xa)) = csqrt(real_Vector_of_real(complex,Xa)) ) ) ).

% of_real_sqrt
tff(fact_6812_quotient__of__denom__pos,axiom,
    ! [R2: rat,P3: int,Q5: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q5) )
     => aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q5) ) ).

% quotient_of_denom_pos
tff(fact_6813_rat__less__code,axiom,
    ! [P3: rat,Q5: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),P3),Q5)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aTP_Lamp_afc(rat,fun(int,fun(int,$o)),Q5)),quotient_of(P3)) ) ).

% rat_less_code
tff(fact_6814_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: 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),Na)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un2480387367778600638or_num(M,Na)) ) ).

% numeral_xor_num
tff(fact_6815_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),pred_numeral(K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_6816_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_6817_Suc__eq__numeral,axiom,
    ! [Na: nat,K: num] :
      ( ( aa(nat,nat,suc,Na) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( Na = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_6818_eq__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,Na) )
    <=> ( pred_numeral(K) = Na ) ) ).

% eq_numeral_Suc
tff(fact_6819_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_6820_less__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),pred_numeral(K)),Na) ) ).

% less_numeral_Suc
tff(fact_6821_less__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_6822_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(bit1(K)) = aa(num,nat,numeral_numeral(nat),bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_6823_le__numeral__Suc,axiom,
    ! [K: num,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Na) ) ).

% le_numeral_Suc
tff(fact_6824_le__Suc__numeral,axiom,
    ! [Na: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_6825_diff__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na)) = aa(nat,nat,minus_minus(nat,pred_numeral(K)),Na) ).

% diff_numeral_Suc
tff(fact_6826_diff__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,minus_minus(nat,Na),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_6827_max__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),Na),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_6828_max__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),Na)) ).

% max_numeral_Suc
tff(fact_6829_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(bit0(K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_6830_min__numeral__Suc,axiom,
    ! [K: num,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Na)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),Na)) ).

% min_numeral_Suc
tff(fact_6831_min__Suc__numeral,axiom,
    ! [Na: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Na)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),Na),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_6832_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_6833_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_6834_lessThan__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_lessThan(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),pred_numeral(K)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_6835_atMost__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_atMost(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_6836_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: num] :
          ( ( bit_un2480387367778600638or_num(M,Na) = 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),Na)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_6837_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,fun(A,A)),V2: num,Na: nat] :
      aa(nat,A,rec_nat(A,A3,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Na)) = $let(
        pv: nat,
        pv:= pred_numeral(V2),
        aa(A,A,aa(nat,fun(A,A),F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Na)),aa(nat,A,rec_nat(A,A3,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pv),Na))) ) ).

% rec_nat_add_eq_if
tff(fact_6838_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,A),V2: num,Na: nat] : case_nat(A,A3,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V2)),Na)) = aa(nat,A,F2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V2)),Na)) ).

% case_nat_add_eq_if
tff(fact_6839_old_Onat_Osimps_I7_J,axiom,
    ! [A: $tType,F13: A,F24: fun(nat,fun(A,A)),Nat: nat] : aa(nat,A,rec_nat(A,F13,F24),aa(nat,nat,suc,Nat)) = aa(A,A,aa(nat,fun(A,A),F24,Nat),aa(nat,A,rec_nat(A,F13,F24),Nat)) ).

% old.nat.simps(7)
tff(fact_6840_old_Onat_Osimps_I6_J,axiom,
    ! [A: $tType,F13: A,F24: fun(nat,fun(A,A))] : aa(nat,A,rec_nat(A,F13,F24),zero_zero(nat)) = F13 ).

% old.nat.simps(6)
tff(fact_6841_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,A),V2: num] : case_nat(A,A3,F2,aa(num,nat,numeral_numeral(nat),V2)) = aa(nat,A,F2,pred_numeral(V2)) ).

% case_nat_numeral
tff(fact_6842_rec__nat__numeral,axiom,
    ! [A: $tType,A3: A,F2: fun(nat,fun(A,A)),V2: num] :
      aa(nat,A,rec_nat(A,A3,F2),aa(num,nat,numeral_numeral(nat),V2)) = $let(
        pv: nat,
        pv:= pred_numeral(V2),
        aa(A,A,aa(nat,fun(A,A),F2,pv),aa(nat,A,rec_nat(A,A3,F2),pv)) ) ).

% rec_nat_numeral
tff(fact_6843_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_afd(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_6844_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_sz(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_6845_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F13: A,F24: fun(nat,A),X23: nat] : case_nat(A,F13,F24,aa(nat,nat,suc,X23)) = aa(nat,A,F24,X23) ).

% old.nat.simps(5)
tff(fact_6846_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F13: A,F24: fun(nat,A)] : case_nat(A,F13,F24,zero_zero(nat)) = F13 ).

% old.nat.simps(4)
tff(fact_6847_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F13: B,F24: fun(nat,B),Nat: nat] : aa(B,A,H,case_nat(B,F13,F24,Nat)) = case_nat(A,aa(B,A,H,F13),aa(fun(nat,B),fun(nat,A),aTP_Lamp_afe(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F24),Nat) ).

% nat.case_distrib
tff(fact_6848_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Na)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% less_eq_nat.simps(2)
tff(fact_6849_max__Suc2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,Na)) = case_nat(nat,aa(nat,nat,suc,Na),aTP_Lamp_aff(nat,fun(nat,nat),Na),M) ).

% max_Suc2
tff(fact_6850_max__Suc1,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,Na)),M) = case_nat(nat,aa(nat,nat,suc,Na),aTP_Lamp_afg(nat,fun(nat,nat),Na),M) ).

% max_Suc1
tff(fact_6851_diff__Suc,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Na)) = case_nat(nat,zero_zero(nat),aTP_Lamp_dt(nat,nat),aa(nat,nat,minus_minus(nat,M),Na)) ).

% diff_Suc
tff(fact_6852_min__Suc2,axiom,
    ! [M: nat,Na: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,Na)) = case_nat(nat,zero_zero(nat),aTP_Lamp_afh(nat,fun(nat,nat),Na),M) ).

% min_Suc2
tff(fact_6853_min__Suc1,axiom,
    ! [Na: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,Na)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_afi(nat,fun(nat,nat),Na),M) ).

% min_Suc1
tff(fact_6854_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xa: A,F2: fun(nat,A),Na: nat] :
      case_nat(A,Xa,F2,Na) = $ite(Na = zero_zero(nat),Xa,aa(nat,A,F2,aa(nat,nat,minus_minus(nat,Na),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_6855_old_Orec__nat__def,axiom,
    ! [A: $tType,X2: A,Xa2: fun(nat,fun(A,A)),Xb2: nat] : aa(nat,A,rec_nat(A,X2,Xa2),Xb2) = the(A,rec_set_nat(A,X2,Xa2,Xb2)) ).

% old.rec_nat_def
tff(fact_6856_rec__nat__0__imp,axiom,
    ! [A: $tType,F2: fun(nat,A),F13: A,F24: fun(nat,fun(A,A))] :
      ( ( F2 = rec_nat(A,F13,F24) )
     => ( aa(nat,A,F2,zero_zero(nat)) = F13 ) ) ).

% rec_nat_0_imp
tff(fact_6857_subset__Collect__iff,axiom,
    ! [A: $tType,B2: set(A),A4: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),A4),P)))
      <=> ! [X: A] :
            ( member(A,X,B2)
           => aa(A,$o,P,X) ) ) ) ).

% subset_Collect_iff
tff(fact_6858_subset__CollectI,axiom,
    ! [A: $tType,B2: set(A),A4: set(A),Q: fun(A,$o),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
     => ( ! [X3: A] :
            ( member(A,X3,B2)
           => ( aa(A,$o,Q,X3)
             => aa(A,$o,P,X3) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),B2),Q))),aa(fun(A,$o),set(A),collect(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) ) ) ).

% subset_CollectI
tff(fact_6859_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P: fun(A,$o),F13: A,F24: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P,case_nat(A,F13,F24,Nat))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ aa(A,$o,P,F13) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ aa(A,$o,P,aa(nat,A,F24,pred(Nat))) ) ) ) ).

% nat.split_sels(2)
tff(fact_6860_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),F13: A,F24: fun(nat,A),Nat: nat] :
      ( aa(A,$o,P,case_nat(A,F13,F24,Nat))
    <=> ( ( ( Nat = zero_zero(nat) )
         => aa(A,$o,P,F13) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => aa(A,$o,P,aa(nat,A,F24,pred(Nat))) ) ) ) ).

% nat.split_sels(1)
tff(fact_6861_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_dt(nat,nat),Nat) ).

% pred_def
tff(fact_6862_relImage__proj,axiom,
    ! [A: $tType,A4: set(A),R: set(product_prod(A,A))] :
      ( equiv_equiv(A,A4,R)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),bNF_Gr4221423524335903396lImage(A,set(A),R,equiv_proj(A,A,R))),id_on(set(A),equiv_quotient(A,A4,R))) ) ).

% relImage_proj
tff(fact_6863_relInvImage__Id__on,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),B2: set(B)] :
      ( ! [A1: A,A22: A] :
          ( ( aa(A,B,F2,A1) = aa(A,B,F2,A22) )
        <=> ( A1 = A22 ) )
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),bNF_Gr7122648621184425601vImage(A,B,A4,id_on(B,B2),F2)),id2(A)) ) ).

% relInvImage_Id_on
tff(fact_6864_relImage__mono,axiom,
    ! [B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),F2: fun(A,B)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R1),R22)
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),bNF_Gr4221423524335903396lImage(A,B,R1,F2)),bNF_Gr4221423524335903396lImage(A,B,R22,F2)) ) ).

% relImage_mono
tff(fact_6865_relInvImage__mono,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),A4: set(B),F2: fun(B,A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R1),R22)
     => aa(set(product_prod(B,B)),$o,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),$o),ord_less_eq(set(product_prod(B,B))),bNF_Gr7122648621184425601vImage(B,A,A4,R1,F2)),bNF_Gr7122648621184425601vImage(B,A,A4,R22,F2)) ) ).

% relInvImage_mono
tff(fact_6866_relInvImage__UNIV__relImage,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F2: fun(A,B)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),bNF_Gr7122648621184425601vImage(A,B,top_top(set(A)),bNF_Gr4221423524335903396lImage(A,B,R,F2),F2)) ).

% relInvImage_UNIV_relImage
tff(fact_6867_ran__map__add,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( ran(A,B,map_add(A,B,M1,M22)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),ran(A,B,M1)),ran(A,B,M22)) ) ) ).

% ran_map_add
tff(fact_6868_set__rec,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = rec_list(set(A),A,bot_bot(set(A)),aTP_Lamp_afj(A,fun(list(A),fun(set(A),set(A)))),Xs) ).

% set_rec
tff(fact_6869_dom__map__add,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Na: fun(A,option(B))] : dom(A,B,map_add(A,B,M,Na)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),dom(A,B,Na)),dom(A,B,M)) ).

% dom_map_add
tff(fact_6870_map__add__comm,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( map_add(A,B,M1,M22) = map_add(A,B,M22,M1) ) ) ).

% map_add_comm
tff(fact_6871_graph__map__add,axiom,
    ! [B: $tType,A: $tType,M1: fun(A,option(B)),M22: fun(A,option(B))] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),dom(A,B,M1)),dom(A,B,M22)) = bot_bot(set(A)) )
     => ( graph(A,B,map_add(A,B,M1,M22)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),graph(A,B,M1)),graph(A,B,M22)) ) ) ).

% graph_map_add
tff(fact_6872_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: 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),Na)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_un7362597486090784418nd_num(M,Na)) ) ).

% numeral_and_num
tff(fact_6873_finite__graph__iff__finite__dom,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),graph(A,B,M))
    <=> aa(set(A),$o,finite_finite2(A),dom(A,B,M)) ) ).

% finite_graph_iff_finite_dom
tff(fact_6874_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Na: num] :
          ( ( bit_un7362597486090784418nd_num(M,Na) = 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),Na)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_6875_connected__closedD,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),A4: set(A),B2: set(A)] :
          ( topolo1966860045006549960nected(A,S3)
         => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2)),S3) = bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
             => ( topolo7761053866217962861closed(A,A4)
               => ( topolo7761053866217962861closed(A,B2)
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),S3) = bot_bot(set(A)) )
                    | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B2),S3) = bot_bot(set(A)) ) ) ) ) ) ) ) ) ).

% connected_closedD
tff(fact_6876_connected__closed,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A)] :
          ( topolo1966860045006549960nected(A,S3)
        <=> ~ ? [A8: set(A),B11: set(A)] :
                ( topolo7761053866217962861closed(A,A8)
                & topolo7761053866217962861closed(A,B11)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A8),B11))
                & ( 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)),A8),B11)),S3) = bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),S3) != bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B11),S3) != bot_bot(set(A)) ) ) ) ) ).

% connected_closed
tff(fact_6877_connected__contains__Ioo,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A4: set(A),A3: A,B3: A] :
          ( topolo1966860045006549960nected(A,A4)
         => ( member(A,A3,A4)
           => ( member(A,B3,A4)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A3,B3)),A4) ) ) ) ) ).

% connected_contains_Ioo
tff(fact_6878_connectedD__interval,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [U2: set(A),Xa: A,Ya: A,Z: A] :
          ( topolo1966860045006549960nected(A,U2)
         => ( member(A,Xa,U2)
           => ( member(A,Ya,U2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Z)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Ya)
                 => member(A,Z,U2) ) ) ) ) ) ) ).

% connectedD_interval
tff(fact_6879_connectedI__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U2: set(A)] :
          ( ! [X3: A,Y: A,Z2: A] :
              ( member(A,X3,U2)
             => ( member(A,Y,U2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Z2)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Y)
                   => member(A,Z2,U2) ) ) ) )
         => topolo1966860045006549960nected(A,U2) ) ) ).

% connectedI_interval
tff(fact_6880_connected__iff__interval,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [U2: set(A)] :
          ( topolo1966860045006549960nected(A,U2)
        <=> ! [X: A] :
              ( member(A,X,U2)
             => ! [Xa3: A] :
                  ( member(A,Xa3,U2)
                 => ! [Z5: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Z5)
                     => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z5),Xa3)
                       => member(A,Z5,U2) ) ) ) ) ) ) ).

% connected_iff_interval
tff(fact_6881_connected__contains__Icc,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A4: set(A),A3: A,B3: A] :
          ( topolo1966860045006549960nected(A,A4)
         => ( member(A,A3,A4)
           => ( member(A,B3,A4)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A3,B3)),A4) ) ) ) ) ).

% connected_contains_Icc
tff(fact_6882_connected__empty,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => topolo1966860045006549960nected(A,bot_bot(set(A))) ) ).

% connected_empty
tff(fact_6883_connected__sing,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Xa: A] : topolo1966860045006549960nected(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) ) ).

% connected_sing
tff(fact_6884_connected__Un,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),Ta: set(A)] :
          ( topolo1966860045006549960nected(A,S3)
         => ( topolo1966860045006549960nected(A,Ta)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S3),Ta) != bot_bot(set(A)) )
             => topolo1966860045006549960nected(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S3),Ta)) ) ) ) ) ).

% connected_Un
tff(fact_6885_connected__Union,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(set(A))] :
          ( ! [S4: set(A)] :
              ( member(set(A),S4,S)
             => topolo1966860045006549960nected(A,S4) )
         => ( ( aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),S) != bot_bot(set(A)) )
           => topolo1966860045006549960nected(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),S)) ) ) ) ).

% connected_Union
tff(fact_6886_not__in__connected__cases,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S: set(A),Xa: A] :
          ( topolo1966860045006549960nected(A,S)
         => ( ~ member(A,Xa,S)
           => ( ( S != bot_bot(set(A)) )
             => ( ( condit941137186595557371_above(A,S)
                 => ~ ! [Y2: A] :
                        ( member(A,Y2,S)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y2),Xa) ) )
               => ~ ( condit1013018076250108175_below(A,S)
                   => ~ ! [Y2: A] :
                          ( member(A,Y2,S)
                         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),Y2) ) ) ) ) ) ) ) ).

% not_in_connected_cases
tff(fact_6887_connected__diff__open__from__closed,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),Ta: set(A),U: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),Ta)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),U)
           => ( topolo1002775350975398744n_open(A,S3)
             => ( topolo7761053866217962861closed(A,Ta)
               => ( topolo1966860045006549960nected(A,U)
                 => ( topolo1966860045006549960nected(A,aa(set(A),set(A),minus_minus(set(A),Ta),S3))
                   => topolo1966860045006549960nected(A,aa(set(A),set(A),minus_minus(set(A),U),S3)) ) ) ) ) ) ) ) ).

% connected_diff_open_from_closed
tff(fact_6888_connectedD,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A4: set(A),U2: set(A),V: set(A)] :
          ( topolo1966860045006549960nected(A,A4)
         => ( topolo1002775350975398744n_open(A,U2)
           => ( topolo1002775350975398744n_open(A,V)
             => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),V)),A4) = bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),U2),V))
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),A4) = bot_bot(set(A)) )
                    | ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),V),A4) = bot_bot(set(A)) ) ) ) ) ) ) ) ) ).

% connectedD
tff(fact_6889_connectedI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U2: set(A)] :
          ( ! [A7: set(A)] :
              ( topolo1002775350975398744n_open(A,A7)
             => ! [B4: set(A)] :
                  ( topolo1002775350975398744n_open(A,B4)
                 => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),U2) != bot_bot(set(A)) )
                   => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B4),U2) != bot_bot(set(A)) )
                     => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),B4)),U2) = bot_bot(set(A)) )
                       => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),U2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A7),B4)) ) ) ) ) )
         => topolo1966860045006549960nected(A,U2) ) ) ).

% connectedI
tff(fact_6890_connected__def,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( topolo1966860045006549960nected(A,S)
        <=> ~ ? [A8: set(A),B11: set(A)] :
                ( topolo1002775350975398744n_open(A,A8)
                & topolo1002775350975398744n_open(A,B11)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A8),B11))
                & ( 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)),A8),B11)),S) = bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A8),S) != bot_bot(set(A)) )
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B11),S) != bot_bot(set(A)) ) ) ) ) ).

% connected_def
tff(fact_6891_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys2: list(B),R2: set(product_prod(A,B))] :
      ( member(product_prod(list(A),list(B)),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys2),listrel(A,B,R2))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [N: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
           => member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),N)),aa(nat,B,nth(B,Ys2),N)),R2) ) ) ) ).

% listrel_iff_nth
tff(fact_6892_disjnt__equiv__class,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( equiv_equiv(A,A4,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A))))),aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))))
      <=> ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2) ) ) ).

% disjnt_equiv_class
tff(fact_6893_disjnt__self__iff__empty,axiom,
    ! [A: $tType,S: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),S),S)
    <=> ( S = bot_bot(set(A)) ) ) ).

% disjnt_self_iff_empty
tff(fact_6894_disjnt__insert1,axiom,
    ! [A: $tType,A3: A,X4: set(A),Y3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X4)),Y3)
    <=> ( ~ member(A,A3,Y3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X4),Y3) ) ) ).

% disjnt_insert1
tff(fact_6895_disjnt__insert2,axiom,
    ! [A: $tType,Y3: set(A),A3: A,X4: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Y3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),X4))
    <=> ( ~ member(A,A3,Y3)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Y3),X4) ) ) ).

% disjnt_insert2
tff(fact_6896_disjnt__Un2,axiom,
    ! [A: $tType,C2: set(A),A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C2),A4)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),C2),B2) ) ) ).

% disjnt_Un2
tff(fact_6897_disjnt__Un1,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),C2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)),C2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),C2)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),B2),C2) ) ) ).

% disjnt_Un1
tff(fact_6898_disjnt__sym,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B2)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),B2),A4) ) ).

% disjnt_sym
tff(fact_6899_disjnt__iff,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B2)
    <=> ! [X: A] :
          ~ ( member(A,X,A4)
            & member(A,X,B2) ) ) ).

% disjnt_iff
tff(fact_6900_disjnt__insert,axiom,
    ! [A: $tType,Xa: A,N4: set(A),M5: set(A)] :
      ( ~ member(A,Xa,N4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),M5),N4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),M5)),N4) ) ) ).

% disjnt_insert
tff(fact_6901_disjnt__empty1,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),bot_bot(set(A))),A4) ).

% disjnt_empty1
tff(fact_6902_disjnt__empty2,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),bot_bot(set(A))) ).

% disjnt_empty2
tff(fact_6903_disjnt__def,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B2)
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2) = bot_bot(set(A)) ) ) ).

% disjnt_def
tff(fact_6904_disjnt__subset1,axiom,
    ! [A: $tType,X4: set(A),Y3: set(A),Z6: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X4),Y3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z6),X4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),Z6),Y3) ) ) ).

% disjnt_subset1
tff(fact_6905_disjnt__subset2,axiom,
    ! [A: $tType,X4: set(A),Y3: set(A),Z6: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X4),Y3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z6),Y3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X4),Z6) ) ) ).

% disjnt_subset2
tff(fact_6906_listrel__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S3)
     => aa(set(product_prod(list(A),list(B))),$o,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),$o),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R2)),listrel(A,B,S3)) ) ).

% listrel_mono
tff(fact_6907_disjnt__ge__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y3: set(A),X4: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),Y3)
         => ( ! [X3: A] :
                ( member(A,X3,X4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),Y3)),X3) )
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),X4),Y3) ) ) ) ).

% disjnt_ge_max
tff(fact_6908_card__Un__disjnt,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,finite_finite2(A),B2)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B2)
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(A),nat,finite_card(A),B2)) ) ) ) ) ).

% card_Un_disjnt
tff(fact_6909_listrel__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R2)),transitive_rtrancl(list(A),listrel1(A,R2))) ).

% listrel_subset_rtrancl_listrel1
tff(fact_6910_listrel__Cons,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),Xa: B,Xs: list(B)] : aa(set(list(B)),set(list(A)),image(list(B),list(A),listrel(B,A,R2)),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),aa(list(B),list(B),cons(B,Xa),Xs)),bot_bot(set(list(B))))) = set_Cons(A,aa(set(B),set(A),image(B,A,R2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B)))),aa(set(list(B)),set(list(A)),image(list(B),list(A),listrel(B,A,R2)),aa(set(list(B)),set(list(B)),aa(list(B),fun(set(list(B)),set(list(B))),insert2(list(B)),Xs),bot_bot(set(list(B)))))) ).

% listrel_Cons
tff(fact_6911_sum__card__image,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( pairwise(A,aTP_Lamp_afk(fun(A,set(B)),fun(A,fun(A,$o)),F2),A4)
       => ( aa(set(set(B)),nat,aa(fun(set(B),nat),fun(set(set(B)),nat),groups7311177749621191930dd_sum(set(B),nat),finite_card(B)),aa(set(A),set(set(B)),image2(A,set(B),F2),A4)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_lv(fun(A,set(B)),fun(A,nat),F2)),A4) ) ) ) ).

% sum_card_image
tff(fact_6912_pairwise__mono,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),A4: set(A),Q: fun(A,fun(A,$o)),B2: set(A)] :
      ( pairwise(A,P,A4)
     => ( ! [X3: A,Y: A] :
            ( aa(A,$o,aa(A,fun(A,$o),P,X3),Y)
           => aa(A,$o,aa(A,fun(A,$o),Q,X3),Y) )
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
         => pairwise(A,Q,B2) ) ) ) ).

% pairwise_mono
tff(fact_6913_pairwise__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),S: set(A),T3: set(A)] :
      ( pairwise(A,P,S)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),S)
       => pairwise(A,P,T3) ) ) ).

% pairwise_subset
tff(fact_6914_pairwise__imageI,axiom,
    ! [B: $tType,A: $tType,A4: set(A),F2: fun(A,B),P: fun(B,fun(B,$o))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,A4)
         => ( member(A,Y,A4)
           => ( ( X3 != Y )
             => ( ( aa(A,B,F2,X3) != aa(A,B,F2,Y) )
               => aa(B,$o,aa(B,fun(B,$o),P,aa(A,B,F2,X3)),aa(A,B,F2,Y)) ) ) ) )
     => pairwise(B,P,aa(set(A),set(B),image2(A,B,F2),A4)) ) ).

% pairwise_imageI
tff(fact_6915_pairwise__image,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(A,$o)),F2: fun(B,A),S3: set(B)] :
      ( pairwise(A,R2,aa(set(B),set(A),image2(B,A,F2),S3))
    <=> pairwise(B,aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_afl(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),R2),F2),S3) ) ).

% pairwise_image
tff(fact_6916_pairwise__empty,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : pairwise(A,P,bot_bot(set(A))) ).

% pairwise_empty
tff(fact_6917_pairwise__insert,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: A,S3: set(A)] :
      ( pairwise(A,R2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),S3))
    <=> ( ! [Y4: A] :
            ( ( member(A,Y4,S3)
              & ( Y4 != Xa ) )
           => ( aa(A,$o,aa(A,fun(A,$o),R2,Xa),Y4)
              & aa(A,$o,aa(A,fun(A,$o),R2,Y4),Xa) ) )
        & pairwise(A,R2,S3) ) ) ).

% pairwise_insert
tff(fact_6918_pairwiseD,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S: set(A),Xa: A,Ya: A] :
      ( pairwise(A,R,S)
     => ( member(A,Xa,S)
       => ( member(A,Ya,S)
         => ( ( Xa != Ya )
           => aa(A,$o,aa(A,fun(A,$o),R,Xa),Ya) ) ) ) ) ).

% pairwiseD
tff(fact_6919_pairwiseI,axiom,
    ! [A: $tType,S: set(A),R: fun(A,fun(A,$o))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,S)
         => ( member(A,Y,S)
           => ( ( X3 != Y )
             => aa(A,$o,aa(A,fun(A,$o),R,X3),Y) ) ) )
     => pairwise(A,R,S) ) ).

% pairwiseI
tff(fact_6920_pairwise__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S: set(A)] :
      ( pairwise(A,R,S)
    <=> ! [X: A] :
          ( member(A,X,S)
         => ! [Xa3: A] :
              ( member(A,Xa3,S)
             => ( ( X != Xa3 )
               => aa(A,$o,aa(A,fun(A,$o),R,X),Xa3) ) ) ) ) ).

% pairwise_def
tff(fact_6921_pairwise__trivial,axiom,
    ! [A: $tType,I5: set(A)] : pairwise(A,aTP_Lamp_adi(A,fun(A,$o)),I5) ).

% pairwise_trivial
tff(fact_6922_pairwise__singleton,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),A4: A] : pairwise(A,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A4),bot_bot(set(A)))) ).

% pairwise_singleton
tff(fact_6923_pairwise__alt,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),S: set(A)] :
      ( pairwise(A,R,S)
    <=> ! [X: A] :
          ( member(A,X,S)
         => ! [Xa3: A] :
              ( member(A,Xa3,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))
             => aa(A,$o,aa(A,fun(A,$o),R,X),Xa3) ) ) ) ).

% pairwise_alt
tff(fact_6924_disjoint__image__subset,axiom,
    ! [A: $tType,A18: set(set(A)),F2: fun(set(A),set(A))] :
      ( pairwise(set(A),disjnt(A),A18)
     => ( ! [X6: set(A)] :
            ( member(set(A),X6,A18)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),F2,X6)),X6) )
       => pairwise(set(A),disjnt(A),aa(set(set(A)),set(set(A)),image2(set(A),set(A),F2),A18)) ) ) ).

% disjoint_image_subset
tff(fact_6925_card__Union__disjoint,axiom,
    ! [A: $tType,C2: set(set(A))] :
      ( pairwise(set(A),disjnt(A),C2)
     => ( ! [A7: set(A)] :
            ( member(set(A),A7,C2)
           => aa(set(A),$o,finite_finite2(A),A7) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2)) = aa(set(set(A)),nat,aa(fun(set(A),nat),fun(set(set(A)),nat),groups7311177749621191930dd_sum(set(A),nat),finite_card(A)),C2) ) ) ) ).

% card_Union_disjoint
tff(fact_6926_infinite__infinite__partition,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ~ ! [C7: fun(nat,set(A))] :
            ( pairwise(nat,aTP_Lamp_afm(fun(nat,set(A)),fun(nat,fun(nat,$o)),C7),top_top(set(nat)))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image2(nat,set(A),C7),top_top(set(nat))))),A4)
             => ~ ! [I3: nat] : ~ aa(set(A),$o,finite_finite2(A),aa(nat,set(A),C7,I3)) ) ) ) ).

% infinite_infinite_partition
tff(fact_6927_listrel1__subset__listrel,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),R4)
     => ( refl_on(A,top_top(set(A)),R4)
       => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R2)),listrel(A,A,R4)) ) ) ).

% listrel1_subset_listrel
tff(fact_6928_lists__empty,axiom,
    ! [A: $tType] : lists(A,bot_bot(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert2(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% lists_empty
tff(fact_6929_refl__on__empty,axiom,
    ! [A: $tType] : refl_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% refl_on_empty
tff(fact_6930_refl__on__Int,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),B2: set(A),S3: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
     => ( refl_on(A,B2,S3)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),S3)) ) ) ).

% refl_on_Int
tff(fact_6931_refl__onD2,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),Xa: A,Ya: A] :
      ( refl_on(A,A4,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2)
       => member(A,Ya,A4) ) ) ).

% refl_onD2
tff(fact_6932_refl__onD1,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),Xa: A,Ya: A] :
      ( refl_on(A,A4,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2)
       => member(A,Xa,A4) ) ) ).

% refl_onD1
tff(fact_6933_refl__onD,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,A4,R2)
     => ( member(A,A3,A4)
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),A3),R2) ) ) ).

% refl_onD
tff(fact_6934_refl__on__Un,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),B2: set(A),S3: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
     => ( refl_on(A,B2,S3)
       => refl_on(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S3)) ) ) ).

% refl_on_Un
tff(fact_6935_refl__Id,axiom,
    ! [A: $tType] : refl_on(A,top_top(set(A)),id2(A)) ).

% refl_Id
tff(fact_6936_refl__on__Id__on,axiom,
    ! [A: $tType,A4: set(A)] : refl_on(A,A4,id_on(A,A4)) ).

% refl_on_Id_on
tff(fact_6937_lists__eq__set,axiom,
    ! [A: $tType,A4: set(A)] : lists(A,A4) = aa(fun(list(A),$o),set(list(A)),collect(list(A)),aTP_Lamp_afn(set(A),fun(list(A),$o),A4)) ).

% lists_eq_set
tff(fact_6938_refl__on__def_H,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
    <=> ( ! [X: product_prod(A,A)] :
            ( member(product_prod(A,A),X,R2)
           => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_afo(set(A),fun(A,fun(A,$o)),A4)),X) )
        & ! [X: A] :
            ( member(A,X,A4)
           => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),X),R2) ) ) ) ).

% refl_on_def'
tff(fact_6939_lists__mono,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => aa(set(list(A)),$o,aa(set(list(A)),fun(set(list(A)),$o),ord_less_eq(set(list(A))),lists(A,A4)),lists(A,B2)) ) ).

% lists_mono
tff(fact_6940_Collect__finite__eq__lists,axiom,
    ! [A: $tType] : aa(fun(set(A),$o),set(set(A)),collect(set(A)),finite_finite2(A)) = aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),lists(A,top_top(set(A)))) ).

% Collect_finite_eq_lists
tff(fact_6941_Collect__finite__subset__eq__lists,axiom,
    ! [A: $tType,T3: set(A)] : aa(fun(set(A),$o),set(set(A)),collect(set(A)),aTP_Lamp_aaz(set(A),fun(set(A),$o),T3)) = aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),lists(A,T3)) ).

% Collect_finite_subset_eq_lists
tff(fact_6942_refl__on__UNION,axiom,
    ! [B: $tType,A: $tType,S: set(A),A4: fun(A,set(B)),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [X3: A] :
          ( member(A,X3,S)
         => refl_on(B,aa(A,set(B),A4,X3),aa(A,set(product_prod(B,B)),R2,X3)) )
     => refl_on(B,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),S)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).

% refl_on_UNION
tff(fact_6943_refl__on__INTER,axiom,
    ! [B: $tType,A: $tType,S: set(A),A4: fun(A,set(B)),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [X3: A] :
          ( member(A,X3,S)
         => refl_on(B,aa(A,set(B),A4,X3),aa(A,set(product_prod(B,B)),R2,X3)) )
     => refl_on(B,aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image2(A,set(B),A4),S)),aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).

% refl_on_INTER
tff(fact_6944_refl__on__singleton,axiom,
    ! [A: $tType,Xa: A] : refl_on(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Xa)),bot_bot(set(product_prod(A,A))))) ).

% refl_on_singleton
tff(fact_6945_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( antisym(A,R2)
       => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
           => ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) )
            <=> ( A3 = B3 ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
tff(fact_6946_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I),J)
     => ( last(nat,upt(I,J)) = aa(nat,nat,minus_minus(nat,J),one_one(nat)) ) ) ).

% last_upt
tff(fact_6947_last__replicate,axiom,
    ! [A: $tType,Na: nat,Xa: A] :
      ( ( Na != zero_zero(nat) )
     => ( last(A,replicate(A,Na,Xa)) = Xa ) ) ).

% last_replicate
tff(fact_6948_last__drop,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),aa(list(A),nat,size_size(list(A)),Xs))
     => ( last(A,drop(A,Na,Xs)) = last(A,Xs) ) ) ).

% last_drop
tff(fact_6949_antisym__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S3)
     => ( antisym(A,S3)
       => antisym(A,R2) ) ) ).

% antisym_subset
tff(fact_6950_antisym__Id__on,axiom,
    ! [A: $tType,A4: set(A)] : antisym(A,id_on(A,A4)) ).

% antisym_Id_on
tff(fact_6951_antisym__Id,axiom,
    ! [A: $tType] : antisym(A,id2(A)) ).

% antisym_Id
tff(fact_6952_antisym__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisym(A,R2)
    <=> ! [X: A,Y4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),X),R2)
           => ( X = Y4 ) ) ) ) ).

% antisym_def
tff(fact_6953_antisymI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X3: A,Y: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),X3),R2)
           => ( X3 = Y ) ) )
     => antisym(A,R2) ) ).

% antisymI
tff(fact_6954_antisymD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( antisym(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),A3),R2)
         => ( A3 = B3 ) ) ) ) ).

% antisymD
tff(fact_6955_antisym__empty,axiom,
    ! [A: $tType] : antisym(A,bot_bot(set(product_prod(A,A)))) ).

% antisym_empty
tff(fact_6956_antisym__singleton,axiom,
    ! [A: $tType,Xa: product_prod(A,A)] : antisym(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),Xa),bot_bot(set(product_prod(A,A))))) ).

% antisym_singleton
tff(fact_6957_subset__subseqs,axiom,
    ! [A: $tType,X4: set(A),Xs: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(list(A),set(A),set2(A),Xs))
     => member(set(A),X4,aa(set(list(A)),set(set(A)),image2(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% subset_subseqs
tff(fact_6958_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Na: nat] :
          ( aa(nat,$o,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),Na)
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),bit0(one2))),Na) != zero_zero(A) )
            & ~ aa(nat,$o,bit_se5641148757651400278ts_bit(A,A3),Na) ) ) ) ).

% bit_not_iff_eq
tff(fact_6959_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_6960_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_6961_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_6962_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xa)),Xa) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_6963_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_6964_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_6965_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Na: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Na))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_6966_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xa: A,Ya: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xa),Ya) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xa),Ya) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xa) = Ya ) ) ) ) ).

% bit.compl_unique
tff(fact_6967_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_6968_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_6969_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One) ) ).

% abstract_boolean_algebra_sym_diff.axioms(1)
tff(fact_6970_abstract__boolean__algebra_Ocompl__one,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,One) = Zero ) ) ).

% abstract_boolean_algebra.compl_one
tff(fact_6971_abstract__boolean__algebra_Ocompl__zero,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,Zero) = One ) ) ).

% abstract_boolean_algebra.compl_zero
tff(fact_6972_abstract__boolean__algebra_Ocompl__unique,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A,Ya: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,aa(A,fun(A,A),Conj,Xa),Ya) = Zero )
       => ( ( aa(A,A,aa(A,fun(A,A),Disj,Xa),Ya) = One )
         => ( aa(A,A,Compl,Xa) = Ya ) ) ) ) ).

% abstract_boolean_algebra.compl_unique
tff(fact_6973_abstract__boolean__algebra_Odouble__compl,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,Compl,Xa)) = Xa ) ) ).

% abstract_boolean_algebra.double_compl
tff(fact_6974_abstract__boolean__algebra_Odisj__one__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,One),Xa) = One ) ) ).

% abstract_boolean_algebra.disj_one_left
tff(fact_6975_abstract__boolean__algebra_Oconj__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Xa),One) = Xa ) ) ).

% abstract_boolean_algebra.conj_one_right
tff(fact_6976_abstract__boolean__algebra_Oconj__zero__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Zero),Xa) = Zero ) ) ).

% abstract_boolean_algebra.conj_zero_left
tff(fact_6977_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A,Ya: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Conj,Xa),Ya)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,Xa)),aa(A,A,Compl,Ya)) ) ) ).

% abstract_boolean_algebra.de_Morgan_conj
tff(fact_6978_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A,Ya: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Disj,Xa),Ya)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,Xa)),aa(A,A,Compl,Ya)) ) ) ).

% abstract_boolean_algebra.de_Morgan_disj
tff(fact_6979_abstract__boolean__algebra_Odisj__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,Xa),One) = One ) ) ).

% abstract_boolean_algebra.disj_one_right
tff(fact_6980_abstract__boolean__algebra_Oconj__zero__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Xa),Zero) = Zero ) ) ).

% abstract_boolean_algebra.conj_zero_right
tff(fact_6981_abstract__boolean__algebra_Odisj__zero__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,Xa),Zero) = Xa ) ) ).

% abstract_boolean_algebra.disj_zero_right
tff(fact_6982_abstract__boolean__algebra_Oconj__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,Xa)),Xa) = Zero ) ) ).

% abstract_boolean_algebra.conj_cancel_left
tff(fact_6983_abstract__boolean__algebra_Odisj__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,Xa)),Xa) = One ) ) ).

% abstract_boolean_algebra.disj_cancel_left
tff(fact_6984_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Ya: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),Ya) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Xa),aa(A,A,Compl,Ya))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,Xa)),Ya)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_def
tff(fact_6985_abstract__boolean__algebra_Ocomplement__unique,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,A3: A,Xa: A,Ya: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),Xa) = Zero )
       => ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),Xa) = One )
         => ( ( aa(A,A,aa(A,fun(A,A),Conj,A3),Ya) = Zero )
           => ( ( aa(A,A,aa(A,fun(A,A),Disj,A3),Ya) = One )
             => ( Xa = Ya ) ) ) ) ) ) ).

% abstract_boolean_algebra.complement_unique
tff(fact_6986_abstract__boolean__algebra_Oconj__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Xa),aa(A,A,Compl,Xa)) = Zero ) ) ).

% abstract_boolean_algebra.conj_cancel_right
tff(fact_6987_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A,Ya: A,Z: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Xa),aa(A,A,aa(A,fun(A,A),Disj,Ya),Z)) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Xa),Ya)),aa(A,A,aa(A,fun(A,A),Conj,Xa),Z)) ) ) ).

% abstract_boolean_algebra.conj_disj_distrib
tff(fact_6988_abstract__boolean__algebra_Odisj__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,Xa),aa(A,A,Compl,Xa)) = One ) ) ).

% abstract_boolean_algebra.disj_cancel_right
tff(fact_6989_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A,Ya: A,Z: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,Xa),aa(A,A,aa(A,fun(A,A),Conj,Ya),Z)) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Xa),Ya)),aa(A,A,aa(A,fun(A,A),Disj,Xa),Z)) ) ) ).

% abstract_boolean_algebra.disj_conj_distrib
tff(fact_6990_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Ya: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),Ya) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Xa),Ya)),aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,Compl,Xa)),aa(A,A,Compl,Ya))) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_def2
tff(fact_6991_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),Xa) = Zero ) ) ).

% abstract_boolean_algebra_sym_diff.xor_self
tff(fact_6992_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xa: A,Ya: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( ( aa(A,A,Compl,Xa) = aa(A,A,Compl,Ya) )
      <=> ( Xa = Ya ) ) ) ).

% abstract_boolean_algebra.compl_eq_compl_iff
tff(fact_6993_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Ya: A,Z: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Ya),Z)),Xa) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Ya),Xa)),aa(A,A,aa(A,fun(A,A),Conj,Z),Xa)) ) ) ).

% abstract_boolean_algebra.conj_disj_distrib2
tff(fact_6994_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Ya: A,Z: A,Xa: A] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,Ya),Z)),Xa) = aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Disj,Ya),Xa)),aa(A,A,aa(A,fun(A,A),Disj,Z),Xa)) ) ) ).

% abstract_boolean_algebra.disj_conj_distrib2
tff(fact_6995_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,One),Xa) = aa(A,A,Compl,Xa) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_one_left
tff(fact_6996_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Ya: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),aa(A,A,aa(A,fun(A,A),Xor,Xa),Ya)) = Ya ) ) ).

% abstract_boolean_algebra_sym_diff.xor_left_self
tff(fact_6997_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),One) = aa(A,A,Compl,Xa) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_one_right
tff(fact_6998_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Ya: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,Xa)),Ya) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,Xa),Ya)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_compl_left
tff(fact_6999_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,Compl,Xa)),Xa) = One ) ) ).

% abstract_boolean_algebra_sym_diff.xor_cancel_left
tff(fact_7000_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Ya: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),aa(A,A,Compl,Ya)) = aa(A,A,Compl,aa(A,A,aa(A,fun(A,A),Xor,Xa),Ya)) ) ) ).

% abstract_boolean_algebra_sym_diff.xor_compl_right
tff(fact_7001_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A,Ya: A,Z: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Conj,Xa),aa(A,A,aa(A,fun(A,A),Xor,Ya),Z)) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Xa),Ya)),aa(A,A,aa(A,fun(A,A),Conj,Xa),Z)) ) ) ).

% abstract_boolean_algebra_sym_diff.conj_xor_distrib
tff(fact_7002_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Xor,Xa),aa(A,A,Compl,Xa)) = One ) ) ).

% abstract_boolean_algebra_sym_diff.xor_cancel_right
tff(fact_7003_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A)),Ya: A,Z: A,Xa: A] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => ( aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,aa(A,fun(A,A),Xor,Ya),Z)),Xa) = aa(A,A,aa(A,fun(A,A),Xor,aa(A,A,aa(A,fun(A,A),Conj,Ya),Xa)),aa(A,A,aa(A,fun(A,A),Conj,Z),Xa)) ) ) ).

% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
tff(fact_7004_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => boolea2506097494486148201lgebra(A,inf_inf(A),sup_sup(A),uminus_uminus(A),bot_bot(A),top_top(A)) ) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_7005_abstract__boolean__algebra__sym__diff_Ointro,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
      ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
     => ( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
       => boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor) ) ) ).

% abstract_boolean_algebra_sym_diff.intro
tff(fact_7006_abstract__boolean__algebra__sym__diff__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
    <=> ( boolea2506097494486148201lgebra(A,Conj,Disj,Compl,Zero,One)
        & boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ) ).

% abstract_boolean_algebra_sym_diff_def
tff(fact_7007_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Xor: fun(A,fun(A,A))] :
      ( boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor)
    <=> ! [X: A,Y4: A] : aa(A,A,aa(A,fun(A,A),Xor,X),Y4) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X),aa(A,A,Compl,Y4))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X)),Y4)) ) ).

% abstract_boolean_algebra_sym_diff_axioms_def
tff(fact_7008_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
    ! [A: $tType,Xor: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Conj: fun(A,fun(A,A)),Compl: fun(A,A)] :
      ( ! [X3: A,Y: A] : aa(A,A,aa(A,fun(A,A),Xor,X3),Y) = aa(A,A,aa(A,fun(A,A),Disj,aa(A,A,aa(A,fun(A,A),Conj,X3),aa(A,A,Compl,Y))),aa(A,A,aa(A,fun(A,A),Conj,aa(A,A,Compl,X3)),Y))
     => boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).

% abstract_boolean_algebra_sym_diff_axioms.intro
tff(fact_7009_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
    ! [A: $tType,Conj: fun(A,fun(A,A)),Disj: fun(A,fun(A,A)),Compl: fun(A,A),Zero: A,One: A,Xor: fun(A,fun(A,A))] :
      ( boolea3799213064322606851m_diff(A,Conj,Disj,Compl,Zero,One,Xor)
     => boolea5476839437570043046axioms(A,Conj,Disj,Compl,Xor) ) ).

% abstract_boolean_algebra_sym_diff.axioms(2)
tff(fact_7010_Real_Opositive_Oabs__eq,axiom,
    ! [Xa: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aa(fun(nat,rat),fun(fun(nat,rat),$o),realrel,Xa),Xa)
     => ( aa(real,$o,positive2,real2(Xa))
      <=> ? [R5: rat] :
            ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
            & ? [K3: nat] :
              ! [N: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
               => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,Xa,N)) ) ) ) ) ).

% Real.positive.abs_eq
tff(fact_7011_sqr_Osimps_I3_J,axiom,
    ! [Na: num] : sqr(bit1(Na)) = bit1(bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(Na)),Na))) ).

% sqr.simps(3)
tff(fact_7012_sqr__conv__mult,axiom,
    ! [Xa: num] : sqr(Xa) = aa(num,num,aa(num,fun(num,num),times_times(num),Xa),Xa) ).

% sqr_conv_mult
tff(fact_7013_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_7014_sqr_Osimps_I2_J,axiom,
    ! [Na: num] : sqr(bit0(Na)) = bit0(bit0(sqr(Na))) ).

% sqr.simps(2)
tff(fact_7015_pow_Osimps_I3_J,axiom,
    ! [Xa: num,Ya: num] : pow(Xa,bit1(Ya)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(Xa,Ya))),Xa) ).

% pow.simps(3)
tff(fact_7016_Real_Opositive_Orsp,axiom,
    aa(fun(fun(nat,rat),$o),$o,aa(fun(fun(nat,rat),$o),fun(fun(fun(nat,rat),$o),$o),bNF_rel_fun(fun(nat,rat),fun(nat,rat),$o,$o,realrel,fequal($o)),aTP_Lamp_afp(fun(nat,rat),$o)),aTP_Lamp_afp(fun(nat,rat),$o)) ).

% Real.positive.rsp
tff(fact_7017_pow_Osimps_I1_J,axiom,
    ! [Xa: num] : pow(Xa,one2) = Xa ).

% pow.simps(1)
tff(fact_7018_transfer__rule__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & semiring_numeral(B)
        & monoid_add(A)
        & semiring_numeral(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
             => aa(fun(num,B),$o,aa(fun(num,A),fun(fun(num,B),$o),bNF_rel_fun(num,num,A,B,fequal(num),R),numeral_numeral(A)),numeral_numeral(B)) ) ) ) ) ).

% transfer_rule_numeral
tff(fact_7019_transfer__rule__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ring_1(B)
        & ring_1(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
             => ( aa(fun(B,B),$o,aa(fun(A,A),fun(fun(B,B),$o),bNF_rel_fun(A,B,A,B,R,R),uminus_uminus(A)),uminus_uminus(B))
               => aa(fun(int,B),$o,aa(fun(int,A),fun(fun(int,B),$o),bNF_rel_fun(int,int,A,B,fequal(int),R),ring_1_of_int(A)),ring_1_of_int(B)) ) ) ) ) ) ).

% transfer_rule_of_int
tff(fact_7020_pow_Osimps_I2_J,axiom,
    ! [Xa: num,Ya: num] : pow(Xa,bit0(Ya)) = sqr(pow(Xa,Ya)) ).

% pow.simps(2)
tff(fact_7021_transfer__rule__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( semiring_1(B)
        & semiring_1(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),R,bNF_rel_fun(A,B,A,B,R,R)),plus_plus(A)),plus_plus(B))
             => aa(fun(nat,B),$o,aa(fun(nat,A),fun(fun(nat,B),$o),bNF_rel_fun(nat,nat,A,B,fequal(nat),R),semiring_1_of_nat(A)),semiring_1_of_nat(B)) ) ) ) ) ).

% transfer_rule_of_nat
tff(fact_7022_fun_Orel__mono,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: fun(A,fun(B,$o)),Ra: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R),Ra)
     => aa(fun(fun(C,A),fun(fun(C,B),$o)),$o,aa(fun(fun(C,A),fun(fun(C,B),$o)),fun(fun(fun(C,A),fun(fun(C,B),$o)),$o),ord_less_eq(fun(fun(C,A),fun(fun(C,B),$o))),bNF_rel_fun(C,C,A,B,fequal(C),R)),bNF_rel_fun(C,C,A,B,fequal(C),Ra)) ) ).

% fun.rel_mono
tff(fact_7023_transfer__rule__of__bool,axiom,
    ! [A: $tType,B: $tType] :
      ( ( zero_neq_one(B)
        & zero_neq_one(A) )
     => ! [R: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),R,zero_zero(A)),zero_zero(B))
         => ( aa(B,$o,aa(A,fun(B,$o),R,one_one(A)),one_one(B))
           => aa(fun($o,B),$o,aa(fun($o,A),fun(fun($o,B),$o),bNF_rel_fun($o,$o,A,B,fequal($o),R),zero_neq_one_of_bool(A)),zero_neq_one_of_bool(B)) ) ) ) ).

% transfer_rule_of_bool
tff(fact_7024_less__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less(nat)),ord_less(nat)) ).

% less_natural.rsp
tff(fact_7025_less__integer_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),fequal(int),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o))),ord_less(int)),ord_less(int)) ).

% less_integer.rsp
tff(fact_7026_less__eq__integer_Orsp,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(int,fun(int,$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(int,int,fun(int,$o),fun(int,$o),fequal(int),bNF_rel_fun(int,int,$o,$o,fequal(int),fequal($o))),ord_less_eq(int)),ord_less_eq(int)) ).

% less_eq_integer.rsp
tff(fact_7027_less__eq__natural_Orsp,axiom,
    aa(fun(nat,fun(nat,$o)),$o,aa(fun(nat,fun(nat,$o)),fun(fun(nat,fun(nat,$o)),$o),bNF_rel_fun(nat,nat,fun(nat,$o),fun(nat,$o),fequal(nat),bNF_rel_fun(nat,nat,$o,$o,fequal(nat),fequal($o))),ord_less_eq(nat)),ord_less_eq(nat)) ).

% less_eq_natural.rsp
tff(fact_7028_fun__mono,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,C2: fun(A,fun(B,$o)),A4: fun(A,fun(B,$o)),B2: fun(C,fun(D,$o)),D3: fun(C,fun(D,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),C2),A4)
     => ( aa(fun(C,fun(D,$o)),$o,aa(fun(C,fun(D,$o)),fun(fun(C,fun(D,$o)),$o),ord_less_eq(fun(C,fun(D,$o))),B2),D3)
       => aa(fun(fun(A,C),fun(fun(B,D),$o)),$o,aa(fun(fun(A,C),fun(fun(B,D),$o)),fun(fun(fun(A,C),fun(fun(B,D),$o)),$o),ord_less_eq(fun(fun(A,C),fun(fun(B,D),$o))),bNF_rel_fun(A,B,C,D,A4,B2)),bNF_rel_fun(A,B,C,D,C2,D3)) ) ) ).

% fun_mono
tff(fact_7029_Real_Opositive_Otransfer,axiom,
    aa(fun(real,$o),$o,aa(fun(fun(nat,rat),$o),fun(fun(real,$o),$o),bNF_rel_fun(fun(nat,rat),real,$o,$o,pcr_real,fequal($o)),aTP_Lamp_afp(fun(nat,rat),$o)),positive2) ).

% Real.positive.transfer
tff(fact_7030_less__eq__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_acp(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less_eq(int)) ).

% less_eq_int.transfer
tff(fact_7031_less__int_Otransfer,axiom,
    aa(fun(int,fun(int,$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(int,fun(int,$o)),$o),bNF_rel_fun(product_prod(nat,nat),int,fun(product_prod(nat,nat),$o),fun(int,$o),pcr_int,bNF_rel_fun(product_prod(nat,nat),int,$o,$o,pcr_int,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_act(nat,fun(nat,fun(product_prod(nat,nat),$o))))),ord_less(int)) ).

% less_int.transfer
tff(fact_7032_zero__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),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_7033_int__transfer,axiom,
    aa(fun(nat,int),$o,aa(fun(nat,product_prod(nat,nat)),fun(fun(nat,int),$o),bNF_rel_fun(nat,nat,product_prod(nat,nat),int,fequal(nat),pcr_int),aTP_Lamp_afq(nat,product_prod(nat,nat))),semiring_1_of_nat(int)) ).

% int_transfer
tff(fact_7034_one__int_Otransfer,axiom,
    aa(int,$o,aa(product_prod(nat,nat),fun(int,$o),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_7035_mono__transfer,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType] :
      ( ( order(B)
        & order(D)
        & order(C)
        & order(A) )
     => ! [A4: fun(A,fun(B,$o)),B2: fun(C,fun(D,$o))] :
          ( bi_total(A,B,A4)
         => ( aa(fun(B,fun(B,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(B,fun(B,$o)),$o),bNF_rel_fun(A,B,fun(A,$o),fun(B,$o),A4,bNF_rel_fun(A,B,$o,$o,A4,fequal($o))),ord_less_eq(A)),ord_less_eq(B))
           => ( aa(fun(D,fun(D,$o)),$o,aa(fun(C,fun(C,$o)),fun(fun(D,fun(D,$o)),$o),bNF_rel_fun(C,D,fun(C,$o),fun(D,$o),B2,bNF_rel_fun(C,D,$o,$o,B2,fequal($o))),ord_less_eq(C)),ord_less_eq(D))
             => aa(fun(fun(B,D),$o),$o,aa(fun(fun(A,C),$o),fun(fun(fun(B,D),$o),$o),bNF_rel_fun(fun(A,C),fun(B,D),$o,$o,bNF_rel_fun(A,B,C,D,A4,B2),fequal($o)),order_mono(A,C)),order_mono(B,D)) ) ) ) ) ).

% mono_transfer
tff(fact_7036_arg__min__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(A,B),P: fun(A,$o),A3: A] :
          ( inj_on(A,B,F2,aa(fun(A,$o),set(A),collect(A),P))
         => ( aa(A,$o,P,A3)
           => ( ! [Y: A] :
                  ( aa(A,$o,P,Y)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,A3)),aa(A,B,F2,Y)) )
             => ( lattices_ord_arg_min(A,B,F2,P) = A3 ) ) ) ) ) ).

% arg_min_inj_eq
tff(fact_7037_arg__min__equality,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [P: fun(A,$o),K: A,F2: fun(A,B)] :
          ( aa(A,$o,P,K)
         => ( ! [X3: A] :
                ( aa(A,$o,P,X3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,K)),aa(A,B,F2,X3)) )
           => ( aa(A,B,F2,lattices_ord_arg_min(A,B,F2,P)) = aa(A,B,F2,K) ) ) ) ) ).

% arg_min_equality
tff(fact_7038_arg__min__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,M: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ( aa(A,$o,P,lattices_ord_arg_min(A,nat,M,P))
        & ! [Y2: A] :
            ( aa(A,$o,P,Y2)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,Y2)) ) ) ) ).

% arg_min_nat_lemma
tff(fact_7039_arg__min__nat__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xa: A,M: fun(A,nat)] :
      ( aa(A,$o,P,Xa)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,M,lattices_ord_arg_min(A,nat,M,P))),aa(A,nat,M,Xa)) ) ).

% arg_min_nat_le
tff(fact_7040_arg__minI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [P: fun(A,$o),Xa: A,F2: fun(A,B),Q: fun(A,$o)] :
          ( aa(A,$o,P,Xa)
         => ( ! [Y: A] :
                ( aa(A,$o,P,Y)
               => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y)),aa(A,B,F2,Xa)) )
           => ( ! [X3: A] :
                  ( aa(A,$o,P,X3)
                 => ( ! [Y2: A] :
                        ( aa(A,$o,P,Y2)
                       => ~ aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y2)),aa(A,B,F2,X3)) )
                   => aa(A,$o,Q,X3) ) )
             => aa(A,$o,Q,lattices_ord_arg_min(A,B,F2,P)) ) ) ) ) ).

% arg_minI
tff(fact_7041_arg__min__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),S: set(A)] : lattic7623131987881927897min_on(A,B,F2,S) = lattices_ord_arg_min(A,B,F2,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)) ) ).

% arg_min_on_def
tff(fact_7042_arg__min__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),P: fun(A,$o)] : lattices_ord_arg_min(A,B,F2,P) = fChoice(A,lattic501386751177426532rg_min(A,B,F2,P)) ) ).

% arg_min_def
tff(fact_7043_in__measures_I2_J,axiom,
    ! [A: $tType,Xa: A,Ya: A,F2: fun(A,nat),Fs: list(fun(A,nat))] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),cons(fun(A,nat),F2),Fs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Ya))
        | ( ( aa(A,nat,F2,Xa) = aa(A,nat,F2,Ya) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),measures(A,Fs)) ) ) ) ).

% in_measures(2)
tff(fact_7044_arg__min__natI,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,M: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => aa(A,$o,P,lattices_ord_arg_min(A,nat,M,P)) ) ).

% arg_min_natI
tff(fact_7045_is__arg__min__arg__min__nat,axiom,
    ! [A: $tType,P: fun(A,$o),Xa: A,M: fun(A,nat)] :
      ( aa(A,$o,P,Xa)
     => aa(A,$o,lattic501386751177426532rg_min(A,nat,M,P),lattices_ord_arg_min(A,nat,M,P)) ) ).

% is_arg_min_arg_min_nat
tff(fact_7046_is__arg__min__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F2: fun(A,B),P: fun(A,$o),Xa: A] :
          ( aa(A,$o,lattic501386751177426532rg_min(A,B,F2,P),Xa)
        <=> ( aa(A,$o,P,Xa)
            & ~ ? [Y4: A] :
                  ( aa(A,$o,P,Y4)
                  & aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,F2,Y4)),aa(A,B,F2,Xa)) ) ) ) ) ).

% is_arg_min_def
tff(fact_7047_is__arg__min__antimono,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [F2: fun(A,B),P: fun(A,$o),Xa: A,Ya: A] :
          ( aa(A,$o,lattic501386751177426532rg_min(A,B,F2,P),Xa)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Ya)),aa(A,B,F2,Xa))
           => ( aa(A,$o,P,Ya)
             => aa(A,$o,lattic501386751177426532rg_min(A,B,F2,P),Ya) ) ) ) ) ).

% is_arg_min_antimono
tff(fact_7048_is__arg__min__linorder,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),P: fun(A,$o),Xa: A] :
          ( aa(A,$o,lattic501386751177426532rg_min(A,B,F2,P),Xa)
        <=> ( aa(A,$o,P,Xa)
            & ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,Y4)) ) ) ) ) ).

% is_arg_min_linorder
tff(fact_7049_measures__less,axiom,
    ! [A: $tType,F2: fun(A,nat),Xa: A,Ya: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Ya))
     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),cons(fun(A,nat),F2),Fs))) ) ).

% measures_less
tff(fact_7050_measures__lesseq,axiom,
    ! [A: $tType,F2: fun(A,nat),Xa: A,Ya: A,Fs: list(fun(A,nat))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F2,Xa)),aa(A,nat,F2,Ya))
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),measures(A,Fs))
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),measures(A,aa(list(fun(A,nat)),list(fun(A,nat)),cons(fun(A,nat),F2),Fs))) ) ) ).

% measures_lesseq
tff(fact_7051_ex__is__arg__min__if__finite,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S: set(A),F2: fun(A,B)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ( S != bot_bot(set(A)) )
           => ? [X_1: A] : aa(A,$o,lattic501386751177426532rg_min(A,B,F2,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),S)),X_1) ) ) ) ).

% ex_is_arg_min_if_finite
tff(fact_7052_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( order_7125193373082350890der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( ( aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) = aa(set(A),set(A),image(A,A,R2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))) )
          <=> ( A3 = B3 ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
tff(fact_7053_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,Na: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M),Na),transitive_rtrancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% pred_nat_trancl_eq_le
tff(fact_7054_partial__order__on__empty,axiom,
    ! [A: $tType] : order_7125193373082350890der_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% partial_order_on_empty
tff(fact_7055_less__eq,axiom,
    ! [M: nat,Na: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M),Na),transitive_trancl(nat,pred_nat))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% less_eq
tff(fact_7056_chains__extend,axiom,
    ! [A: $tType,C3: set(set(A)),S: set(set(A)),Z: set(A)] :
      ( member(set(set(A)),C3,chains2(A,S))
     => ( member(set(A),Z,S)
       => ( ! [X3: set(A)] :
              ( member(set(A),X3,C3)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Z) )
         => 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)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Z),bot_bot(set(set(A))))),C3),chains2(A,S)) ) ) ) ).

% chains_extend
tff(fact_7057_inj__on__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),A3: B] :
      ( inj_on(A,B,F2,A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),A4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_afr(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F2),A4),A3))),bot_bot(set(A)))) ) ).

% inj_on_vimage_singleton
tff(fact_7058_vimageI,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: B,B3: A,B2: set(A)] :
      ( ( aa(B,A,F2,A3) = B3 )
     => ( member(A,B3,B2)
       => member(B,A3,vimage(B,A,F2,B2)) ) ) ).

% vimageI
tff(fact_7059_vimage__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F2: fun(A,B),B2: set(B)] :
      ( member(A,A3,vimage(A,B,F2,B2))
    <=> member(B,aa(A,B,F2,A3),B2) ) ).

% vimage_eq
tff(fact_7060_vimage__ident,axiom,
    ! [A: $tType,Y3: set(A)] : vimage(A,A,aTP_Lamp_jq(A,A),Y3) = Y3 ).

% vimage_ident
tff(fact_7061_vimage__Collect__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),P: fun(B,$o)] : vimage(A,B,F2,aa(fun(B,$o),set(B),collect(B),P)) = aa(fun(A,$o),set(A),collect(A),aa(fun(B,$o),fun(A,$o),aTP_Lamp_sp(fun(A,B),fun(fun(B,$o),fun(A,$o)),F2),P)) ).

% vimage_Collect_eq
tff(fact_7062_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_7063_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_7064_vimage__Int,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B),B2: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,A4)),vimage(A,B,F2,B2)) ).

% vimage_Int
tff(fact_7065_vimage__Un,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B),B2: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),vimage(A,B,F2,A4)),vimage(A,B,F2,B2)) ).

% vimage_Un
tff(fact_7066_filtercomap__principal,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B)] : filtercomap(A,B,F2,principal(B,A4)) = principal(A,vimage(A,B,F2,A4)) ).

% filtercomap_principal
tff(fact_7067_vimage__const,axiom,
    ! [B: $tType,A: $tType,C3: B,A4: set(B)] :
      vimage(A,B,aTP_Lamp_ke(B,fun(A,B),C3),A4) = $ite(member(B,C3,A4),top_top(set(A)),bot_bot(set(A))) ).

% vimage_const
tff(fact_7068_image__vimage__eq,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(A)] : aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,A4)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),aa(set(B),set(A),image2(B,A,F2),top_top(set(B)))) ).

% image_vimage_eq
tff(fact_7069_vimage__if,axiom,
    ! [B: $tType,A: $tType,B2: set(A),C3: B,D2: B,A4: set(B)] :
      vimage(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_afs(set(A),fun(B,fun(B,fun(A,B))),B2),C3),D2),A4) = $ite(
        member(B,C3,A4),
        $ite(member(B,D2,A4),top_top(set(A)),B2),
        $ite(member(B,D2,A4),aa(set(A),set(A),uminus_uminus(set(A)),B2),bot_bot(set(A))) ) ).

% vimage_if
tff(fact_7070_finite__vimageI,axiom,
    ! [B: $tType,A: $tType,F3: set(A),H: fun(B,A)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( inj_on(B,A,H,top_top(set(B)))
       => aa(set(B),$o,finite_finite2(B),vimage(B,A,H,F3)) ) ) ).

% finite_vimageI
tff(fact_7071_finite__vimage__Suc__iff,axiom,
    ! [F3: set(nat)] :
      ( aa(set(nat),$o,finite_finite2(nat),vimage(nat,nat,suc,F3))
    <=> aa(set(nat),$o,finite_finite2(nat),F3) ) ).

% finite_vimage_Suc_iff
tff(fact_7072_vimage__inter__cong,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,B),G: fun(A,B),Ya: set(B)] :
      ( ! [W: A] :
          ( member(A,W,S)
         => ( aa(A,B,F2,W) = aa(A,B,G,W) ) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,Ya)),S) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,G,Ya)),S) ) ) ).

% vimage_inter_cong
tff(fact_7073_vimage__Compl,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B)] : vimage(A,B,F2,aa(set(B),set(B),uminus_uminus(set(B)),A4)) = aa(set(A),set(A),uminus_uminus(set(A)),vimage(A,B,F2,A4)) ).

% vimage_Compl
tff(fact_7074_vimageD,axiom,
    ! [A: $tType,B: $tType,A3: A,F2: fun(A,B),A4: set(B)] :
      ( member(A,A3,vimage(A,B,F2,A4))
     => member(B,aa(A,B,F2,A3),A4) ) ).

% vimageD
tff(fact_7075_vimageE,axiom,
    ! [A: $tType,B: $tType,A3: A,F2: fun(A,B),B2: set(B)] :
      ( member(A,A3,vimage(A,B,F2,B2))
     => member(B,aa(A,B,F2,A3),B2) ) ).

% vimageE
tff(fact_7076_vimageI2,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A3: B,A4: set(A)] :
      ( member(A,aa(B,A,F2,A3),A4)
     => member(B,A3,vimage(B,A,F2,A4)) ) ).

% vimageI2
tff(fact_7077_vimage__Collect,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),F2: fun(A,B),Q: fun(A,$o)] :
      ( ! [X3: A] :
          ( aa(B,$o,P,aa(A,B,F2,X3))
        <=> aa(A,$o,Q,X3) )
     => ( vimage(A,B,F2,aa(fun(B,$o),set(B),collect(B),P)) = aa(fun(A,$o),set(A),collect(A),Q) ) ) ).

% vimage_Collect
tff(fact_7078_vimage__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B2: set(B)] : vimage(A,B,F2,B2) = aa(fun(A,$o),set(A),collect(A),aa(set(B),fun(A,$o),aTP_Lamp_vs(fun(A,B),fun(set(B),fun(A,$o)),F2),B2)) ).

% vimage_def
tff(fact_7079_vimage__Diff,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B),B2: set(B)] : vimage(A,B,F2,aa(set(B),set(B),minus_minus(set(B),A4),B2)) = aa(set(A),set(A),minus_minus(set(A),vimage(A,B,F2,A4)),vimage(A,B,F2,B2)) ).

% vimage_Diff
tff(fact_7080_finite__vimage__iff,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F3: set(B)] :
      ( bij_betw(A,B,H,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),$o,finite_finite2(A),vimage(A,B,H,F3))
      <=> aa(set(B),$o,finite_finite2(B),F3) ) ) ).

% finite_vimage_iff
tff(fact_7081_finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F3: set(A),H: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( inj_on(B,A,H,A4)
       => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,F3)),A4)) ) ) ).

% finite_vimage_IntI
tff(fact_7082_vimage__Suc__insert__Suc,axiom,
    ! [Na: nat,A4: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),aa(nat,nat,suc,Na)),A4)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),Na),vimage(nat,nat,suc,A4)) ).

% vimage_Suc_insert_Suc
tff(fact_7083_vimage__singleton__eq,axiom,
    ! [A: $tType,B: $tType,A3: A,F2: fun(A,B),B3: B] :
      ( member(A,A3,vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B3),bot_bot(set(B)))))
    <=> ( aa(A,B,F2,A3) = B3 ) ) ).

% vimage_singleton_eq
tff(fact_7084_vimage__insert,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A3: B,B2: set(B)] : vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),B2)) = 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),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),vimage(A,B,F2,B2)) ).

% vimage_insert
tff(fact_7085_surj__vimage__empty,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => ( ( vimage(B,A,F2,A4) = bot_bot(set(B)) )
      <=> ( A4 = bot_bot(set(A)) ) ) ) ).

% surj_vimage_empty
tff(fact_7086_vimage__image__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] : vimage(A,B,F2,aa(set(A),set(B),image2(A,B,F2),A4)) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aft(fun(A,B),fun(set(A),fun(A,$o)),F2),A4)) ).

% vimage_image_eq
tff(fact_7087_finite__vimageD,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F3: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),vimage(A,B,H,F3))
     => ( ( aa(set(A),set(B),image2(A,B,H),top_top(set(A))) = top_top(set(B)) )
       => aa(set(B),$o,finite_finite2(B),F3) ) ) ).

% finite_vimageD
tff(fact_7088_vimage__Suc__insert__0,axiom,
    ! [A4: set(nat)] : vimage(nat,nat,suc,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),zero_zero(nat)),A4)) = vimage(nat,nat,suc,A4) ).

% vimage_Suc_insert_0
tff(fact_7089_vimage__subsetD,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),B2: set(A),A4: set(B)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),top_top(set(B))) = top_top(set(A)) )
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),vimage(B,A,F2,B2)),A4)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),A4)) ) ) ).

% vimage_subsetD
tff(fact_7090_image__subset__iff__subset__vimage,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),A4)),B2)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),vimage(B,A,F2,B2)) ) ).

% image_subset_iff_subset_vimage
tff(fact_7091_image__vimage__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,F2),vimage(B,A,F2,A4))),A4) ).

% image_vimage_subset
tff(fact_7092_chainsD2,axiom,
    ! [A: $tType,C3: set(set(A)),S: set(set(A))] :
      ( member(set(set(A)),C3,chains2(A,S))
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C3),S) ) ).

% chainsD2
tff(fact_7093_vimage__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),vimage(B,A,F2,A4)),vimage(B,A,F2,B2)) ) ).

% vimage_mono
tff(fact_7094_subset__vimage__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(A,B),B2: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),vimage(A,B,F2,B2))
    <=> ! [X: A] :
          ( member(A,X,A4)
         => member(B,aa(A,B,F2,X),B2) ) ) ).

% subset_vimage_iff
tff(fact_7095_Zorn__Lemma2,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ! [X3: set(set(A))] :
          ( member(set(set(A)),X3,chains2(A,A4))
         => ? [Xa2: set(A)] :
              ( member(set(A),Xa2,A4)
              & ! [Xb: set(A)] :
                  ( member(set(A),Xb,X3)
                 => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xb),Xa2) ) ) )
     => ? [X3: set(A)] :
          ( member(set(A),X3,A4)
          & ! [Xa2: set(A)] :
              ( member(set(A),Xa2,A4)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa2)
               => ( Xa2 = X3 ) ) ) ) ) ).

% Zorn_Lemma2
tff(fact_7096_chainsD,axiom,
    ! [A: $tType,C3: set(set(A)),S: set(set(A)),Xa: set(A),Ya: set(A)] :
      ( member(set(set(A)),C3,chains2(A,S))
     => ( member(set(A),Xa,C3)
       => ( member(set(A),Ya,C3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa),Ya)
            | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ya),Xa) ) ) ) ) ).

% chainsD
tff(fact_7097_Zorn__Lemma,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ! [X3: set(set(A))] :
          ( member(set(set(A)),X3,chains2(A,A4))
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),X3),A4) )
     => ? [X3: set(A)] :
          ( member(set(A),X3,A4)
          & ! [Xa2: set(A)] :
              ( member(set(A),Xa2,A4)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa2)
               => ( Xa2 = X3 ) ) ) ) ) ).

% Zorn_Lemma
tff(fact_7098_continuous__imp__open__vimage,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S3: set(A),F2: fun(A,B),B2: set(B)] :
          ( topolo81223032696312382ous_on(A,B,S3,F2)
         => ( topolo1002775350975398744n_open(A,S3)
           => ( topolo1002775350975398744n_open(B,B2)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,B2)),S3)
               => topolo1002775350975398744n_open(A,vimage(A,B,F2,B2)) ) ) ) ) ) ).

% continuous_imp_open_vimage
tff(fact_7099_dependent__wellorder__choice,axiom,
    ! [A: $tType,B: $tType] :
      ( wellorder(B)
     => ! [P: fun(fun(B,A),fun(B,fun(A,$o)))] :
          ( ! [R3: A,F5: fun(B,A),G7: fun(B,A),X3: B] :
              ( ! [Y2: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y2),X3)
                 => ( aa(B,A,F5,Y2) = aa(B,A,G7,Y2) ) )
             => ( aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,F5),X3),R3)
              <=> aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,G7),X3),R3) ) )
         => ( ! [X3: B,F5: fun(B,A)] :
                ( ! [Y2: B] :
                    ( aa(B,$o,aa(B,fun(B,$o),ord_less(B),Y2),X3)
                   => aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,F5),Y2),aa(B,A,F5,Y2)) )
               => ? [X_12: A] : aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,F5),X3),X_12) )
           => ? [F5: fun(B,A)] :
              ! [X2: B] : aa(A,$o,aa(B,fun(A,$o),aa(fun(B,A),fun(B,fun(A,$o)),P,F5),X2),aa(B,A,F5,X2)) ) ) ) ).

% dependent_wellorder_choice
tff(fact_7100_finite__vimageD_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),vimage(A,B,F2,A4))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(A),set(B),image2(A,B,F2),top_top(set(A))))
       => aa(set(B),$o,finite_finite2(B),A4) ) ) ).

% finite_vimageD'
tff(fact_7101_inf__img__fin__dom,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A4))
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ? [X3: A] :
            ( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A4))
            & ~ aa(set(B),$o,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_dom
tff(fact_7102_inf__img__fin__domE,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A4))
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ~ ! [Y: A] :
              ( member(A,Y,aa(set(B),set(A),image2(B,A,F2),A4))
             => aa(set(B),$o,finite_finite2(B),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))) ) ) ) ).

% inf_img_fin_domE
tff(fact_7103_vimage__subsetI,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B2: set(B),A4: set(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),image2(A,B,F2),A4))
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,B2)),A4) ) ) ).

% vimage_subsetI
tff(fact_7104_finite__finite__vimage__IntI,axiom,
    ! [A: $tType,B: $tType,F3: set(A),H: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),F3)
     => ( ! [Y: A] :
            ( member(A,Y,F3)
           => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),A4)) )
       => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,H,F3)),A4)) ) ) ).

% finite_finite_vimage_IntI
tff(fact_7105_countable__vimage,axiom,
    ! [B: $tType,A: $tType,B2: set(A),F2: fun(B,A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),top_top(set(B))))
     => ( countable_countable(B,vimage(B,A,F2,B2))
       => countable_countable(A,B2) ) ) ).

% countable_vimage
tff(fact_7106_vimage__subset__eq,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),B2: set(B),A4: set(A)] :
      ( bij_betw(A,B,F2,top_top(set(A)),top_top(set(B)))
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,B2)),A4)
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),image2(A,B,F2),A4)) ) ) ).

% vimage_subset_eq
tff(fact_7107_vimage__eq__UN,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),B2: set(B)] : vimage(A,B,F2,B2) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),aTP_Lamp_afu(fun(A,B),fun(B,set(A)),F2)),B2)) ).

% vimage_eq_UN
tff(fact_7108_inf__img__fin__domE_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A4))
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ~ ! [Y: A] :
              ( member(A,Y,aa(set(B),set(A),image2(B,A,F2),A4))
             => aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A))))),A4)) ) ) ) ).

% inf_img_fin_domE'
tff(fact_7109_inf__img__fin__dom_H,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),aa(set(B),set(A),image2(B,A,F2),A4))
     => ( ~ aa(set(B),$o,finite_finite2(B),A4)
       => ? [X3: A] :
            ( member(A,X3,aa(set(B),set(A),image2(B,A,F2),A4))
            & ~ aa(set(B),$o,finite_finite2(B),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),vimage(B,A,F2,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),bot_bot(set(A))))),A4)) ) ) ) ).

% inf_img_fin_dom'
tff(fact_7110_card__vimage__inj,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),A4: set(B)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(A),set(B),image2(A,B,F2),top_top(set(A))))
       => ( aa(set(A),nat,finite_card(A),vimage(A,B,F2,A4)) = aa(set(B),nat,finite_card(B),A4) ) ) ) ).

% card_vimage_inj
tff(fact_7111_card__vimage__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),D3: set(A),A4: set(B)] :
      ( inj_on(A,B,F2,D3)
     => ( aa(set(B),$o,finite_finite2(B),A4)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),vimage(A,B,F2,A4)),D3))),aa(set(B),nat,finite_card(B),A4)) ) ) ).

% card_vimage_inj_on_le
tff(fact_7112_inj__vimage__singleton,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A3: B] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),vimage(A,B,F2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),A3),bot_bot(set(B))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),the(A,aa(B,fun(A,$o),aTP_Lamp_afv(fun(A,B),fun(B,fun(A,$o)),F2),A3))),bot_bot(set(A)))) ) ).

% inj_vimage_singleton
tff(fact_7113_chains__def,axiom,
    ! [A: $tType,A4: set(set(A))] : chains2(A,A4) = aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),aTP_Lamp_afw(set(set(A)),fun(set(set(A)),$o),A4)) ).

% chains_def
tff(fact_7114_image__split__eq__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),A4: set(C)] : aa(set(C),set(product_prod(A,B)),image2(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_afx(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G)),A4) = product_Sigma(A,B,aa(set(C),set(A),image2(C,A,F2),A4),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_afy(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),F2),G),A4)) ).

% image_split_eq_Sigma
tff(fact_7115_Field__square,axiom,
    ! [A: $tType,Xa: set(A)] : aa(set(product_prod(A,A)),set(A),field2(A),product_Sigma(A,A,Xa,aTP_Lamp_afz(set(A),fun(A,set(A)),Xa))) = Xa ).

% Field_square
tff(fact_7116_Sigma__empty1,axiom,
    ! [B: $tType,A: $tType,B2: fun(A,set(B))] : product_Sigma(A,B,bot_bot(set(A)),B2) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty1
tff(fact_7117_Sigma__empty2,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : product_Sigma(A,B,A4,aTP_Lamp_aga(A,set(B))) = bot_bot(set(product_prod(A,B))) ).

% Sigma_empty2
tff(fact_7118_Times__empty,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ( product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)) = bot_bot(set(product_prod(A,B))) )
    <=> ( ( A4 = bot_bot(set(A)) )
        | ( B2 = bot_bot(set(B)) ) ) ) ).

% Times_empty
tff(fact_7119_disjnt__Times1__iff,axiom,
    ! [A: $tType,B: $tType,C2: set(A),A4: set(B),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,C2,aTP_Lamp_agb(set(B),fun(A,set(B)),A4))),product_Sigma(A,B,C2,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))
    <=> ( ( C2 = bot_bot(set(A)) )
        | aa(set(B),$o,aa(set(B),fun(set(B),$o),disjnt(B),A4),B2) ) ) ).

% disjnt_Times1_iff
tff(fact_7120_disjnt__Times2__iff,axiom,
    ! [B: $tType,A: $tType,A4: set(A),C2: set(B),B2: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),C2))),product_Sigma(A,B,B2,aTP_Lamp_agb(set(B),fun(A,set(B)),C2)))
    <=> ( ( C2 = bot_bot(set(B)) )
        | aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B2) ) ) ).

% disjnt_Times2_iff
tff(fact_7121_finite__SigmaI,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [A5: A] :
            ( member(A,A5,A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B2,A5)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,B2)) ) ) ).

% finite_SigmaI
tff(fact_7122_connected__Times__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [S: set(A),T3: set(B)] :
          ( topolo1966860045006549960nected(product_prod(A,B),product_Sigma(A,B,S,aTP_Lamp_agc(set(B),fun(A,set(B)),T3)))
        <=> ( ( S = bot_bot(set(A)) )
            | ( T3 = bot_bot(set(B)) )
            | ( topolo1966860045006549960nected(A,S)
              & topolo1966860045006549960nected(B,T3) ) ) ) ) ).

% connected_Times_eq
tff(fact_7123_insert__Times__insert,axiom,
    ! [A: $tType,B: $tType,A3: A,A4: set(A),B3: B,B2: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4),aa(set(B),fun(A,set(B)),aTP_Lamp_agd(B,fun(set(B),fun(A,set(B))),B3),B2)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3)),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A4,aa(set(B),fun(A,set(B)),aTP_Lamp_agd(B,fun(set(B),fun(A,set(B))),B3),B2))),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4),aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))) ).

% insert_Times_insert
tff(fact_7124_card__SigmaI,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ! [X3: A] :
            ( member(A,X3,A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B2,X3)) )
       => ( aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,A4,B2)) = aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7311177749621191930dd_sum(A,nat),aTP_Lamp_lv(fun(A,set(B)),fun(A,nat),B2)),A4) ) ) ) ).

% card_SigmaI
tff(fact_7125_Pair__vimage__Sigma,axiom,
    ! [B: $tType,A: $tType,Xa: B,A4: set(B),F2: fun(B,set(A))] :
      vimage(A,product_prod(B,A),product_Pair(B,A,Xa),product_Sigma(B,A,A4,F2)) = $ite(member(B,Xa,A4),aa(B,set(A),F2,Xa),bot_bot(set(A))) ).

% Pair_vimage_Sigma
tff(fact_7126_trancl__subset__Sigma,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))) ) ).

% trancl_subset_Sigma
tff(fact_7127_listrel__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))
     => aa(set(product_prod(list(A),list(A))),$o,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),$o),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R2)),product_Sigma(list(A),list(A),lists(A,A4),aTP_Lamp_age(set(A),fun(list(A),set(list(A))),A4))) ) ).

% listrel_subset
tff(fact_7128_Id__on__subset__Times,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),id_on(A,A4)),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))) ).

% Id_on_subset_Times
tff(fact_7129_Restr__subset,axiom,
    ! [A: $tType,A4: set(A),B2: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),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,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2)))),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))) ) ) ).

% Restr_subset
tff(fact_7130_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A4: set(A),C2: set(A),B2: fun(A,set(B)),D3: fun(A,set(B))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),C2)
     => ( ! [X3: A] :
            ( member(A,X3,A4)
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),B2,X3)),aa(A,set(B),D3,X3)) )
       => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A4,B2)),product_Sigma(A,B,C2,D3)) ) ) ).

% Sigma_mono
tff(fact_7131_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,Xa: A,C2: set(A),A4: set(B),B2: set(B)] :
      ( member(A,Xa,C2)
     => ( aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A4,aTP_Lamp_ld(set(A),fun(B,set(A)),C2))),product_Sigma(B,A,B2,aTP_Lamp_ld(set(A),fun(B,set(A)),C2)))
      <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),B2) ) ) ).

% Times_subset_cancel2
tff(fact_7132_equiv__type,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( equiv_equiv(A,A4,R2)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))) ) ).

% equiv_type
tff(fact_7133_times__eq__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B),C2: set(A),D3: set(B)] :
      ( ( product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)) = product_Sigma(A,B,C2,aTP_Lamp_agb(set(B),fun(A,set(B)),D3)) )
    <=> ( ( ( A4 = C2 )
          & ( B2 = D3 ) )
        | ( ( ( A4 = bot_bot(set(A)) )
            | ( B2 = bot_bot(set(B)) ) )
          & ( ( C2 = bot_bot(set(A)) )
            | ( D3 = bot_bot(set(B)) ) ) ) ) ) ).

% times_eq_iff
tff(fact_7134_Sigma__empty__iff,axiom,
    ! [A: $tType,B: $tType,I5: set(A),X4: fun(A,set(B))] :
      ( ( product_Sigma(A,B,I5,X4) = bot_bot(set(product_prod(A,B))) )
    <=> ! [X: A] :
          ( member(A,X,I5)
         => ( aa(A,set(B),X4,X) = bot_bot(set(B)) ) ) ) ).

% Sigma_empty_iff
tff(fact_7135_Times__Un__distrib1,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(A),C2: set(B)] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2),aTP_Lamp_agb(set(B),fun(A,set(B)),C2)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),C2))),product_Sigma(A,B,B2,aTP_Lamp_agb(set(B),fun(A,set(B)),C2))) ).

% Times_Un_distrib1
tff(fact_7136_Sigma__Un__distrib2,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A4: fun(A,set(B)),B2: fun(A,set(B))] : product_Sigma(A,B,I5,aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_agf(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),A4),B2)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I5,A4)),product_Sigma(A,B,I5,B2)) ).

% Sigma_Un_distrib2
tff(fact_7137_Sigma__Un__distrib1,axiom,
    ! [B: $tType,A: $tType,I5: set(A),J4: set(A),C2: fun(A,set(B))] : product_Sigma(A,B,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),I5),J4),C2) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),product_Sigma(A,B,I5,C2)),product_Sigma(A,B,J4,C2)) ).

% Sigma_Un_distrib1
tff(fact_7138_finite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2))) ) ) ).

% finite_cartesian_product
tff(fact_7139_infinite__cartesian__product,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ~ aa(set(B),$o,finite_finite2(B),B2)
       => ~ aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2))) ) ) ).

% infinite_cartesian_product
tff(fact_7140_disjnt__Sigma__iff,axiom,
    ! [B: $tType,A: $tType,A4: set(A),C2: fun(A,set(B)),B2: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),disjnt(product_prod(A,B)),product_Sigma(A,B,A4,C2)),product_Sigma(A,B,B2,C2))
    <=> ( ! [X: A] :
            ( member(A,X,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),B2))
           => ( aa(A,set(B),C2,X) = bot_bot(set(B)) ) )
        | aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),A4),B2) ) ) ).

% disjnt_Sigma_iff
tff(fact_7141_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),C2: set(B),B2: set(A),D3: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),C2))),product_Sigma(A,B,B2,aTP_Lamp_agb(set(B),fun(A,set(B)),D3)))
    <=> ( ( A4 = bot_bot(set(A)) )
        | ( C2 = bot_bot(set(B)) )
        | ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C2),D3) ) ) ) ).

% times_subset_iff
tff(fact_7142_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A3: A,B3: A,R2: set(product_prod(A,A)),A4: set(A)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),transitive_rtrancl(A,R2))
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))
       => ( ( A3 = B3 )
          | member(A,A3,A4) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_7143_Field__Restr__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,A)),set(A),field2(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))))),A4) ).

% Field_Restr_subset
tff(fact_7144_wfI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),B2)))
     => ( ! [X3: A,P6: fun(A,$o)] :
            ( ! [Xa2: A] :
                ( ! [Y: A] :
                    ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),Xa2),R2)
                   => aa(A,$o,P6,Y) )
               => aa(A,$o,P6,Xa2) )
           => ( member(A,X3,A4)
             => ( member(A,X3,B2)
               => aa(A,$o,P6,X3) ) ) )
       => wf(A,R2) ) ) ).

% wfI
tff(fact_7145_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A4: fun(B,set(A)),I5: set(B)] :
    ? [F5: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F5,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5)))
      & aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),aa(set(A),set(product_prod(B,A)),image2(A,product_prod(B,A),F5),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),A4),I5)))),product_Sigma(B,A,I5,A4)) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_7146_finite__cartesian__product__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))
    <=> ( ( A4 = bot_bot(set(A)) )
        | ( B2 = bot_bot(set(B)) )
        | ( aa(set(A),$o,finite_finite2(A),A4)
          & aa(set(B),$o,finite_finite2(B),B2) ) ) ) ).

% finite_cartesian_product_iff
tff(fact_7147_finite__cartesian__productD2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))
     => ( ( A4 != bot_bot(set(A)) )
       => aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% finite_cartesian_productD2
tff(fact_7148_finite__cartesian__productD1,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))
     => ( ( B2 != bot_bot(set(B)) )
       => aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% finite_cartesian_productD1
tff(fact_7149_finite__SigmaI2,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: fun(A,set(B))] :
      ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_agg(set(A),fun(fun(A,set(B)),fun(A,$o)),A4),B2)))
     => ( ! [A5: A] :
            ( member(A,A5,A4)
           => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B2,A5)) )
       => aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),product_Sigma(A,B,A4,B2)) ) ) ).

% finite_SigmaI2
tff(fact_7150_chain__subset__def,axiom,
    ! [A: $tType,C2: set(set(A))] :
      ( chain_subset(A,C2)
    <=> ! [X: set(A)] :
          ( member(set(A),X,C2)
         => ! [Xa3: set(A)] :
              ( member(set(A),Xa3,C2)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),Xa3)
                | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa3),X) ) ) ) ) ).

% chain_subset_def
tff(fact_7151_Image__subset,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),A4: set(A),B2: set(B),C2: set(A)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,R2),C2)),B2) ) ).

% Image_subset
tff(fact_7152_trancl__subset__Field2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_agh(set(product_prod(A,A)),fun(A,set(A)),R2))) ).

% trancl_subset_Field2
tff(fact_7153_refl__on__def,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
    <=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))
        & ! [X: A] :
            ( member(A,X,A4)
           => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),X),R2) ) ) ) ).

% refl_on_def
tff(fact_7154_refl__onI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))
     => ( ! [X3: A] :
            ( member(A,X3,A4)
           => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),X3),R2) )
       => refl_on(A,A4,R2) ) ) ).

% refl_onI
tff(fact_7155_finite__equiv__class,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),X4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))
       => ( member(set(A),X4,equiv_quotient(A,A4,R2))
         => aa(set(A),$o,finite_finite2(A),X4) ) ) ) ).

% finite_equiv_class
tff(fact_7156_open__prod__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B))] :
          ( ! [X3: product_prod(A,B)] :
              ( member(product_prod(A,B),X3,S)
             => ? [A19: set(A),B8: set(B)] :
                  ( topolo1002775350975398744n_open(A,A19)
                  & topolo1002775350975398744n_open(B,B8)
                  & member(product_prod(A,B),X3,product_Sigma(A,B,A19,aTP_Lamp_agc(set(B),fun(A,set(B)),B8)))
                  & aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A19,aTP_Lamp_agc(set(B),fun(A,set(B)),B8))),S) ) )
         => topolo1002775350975398744n_open(product_prod(A,B),S) ) ) ).

% open_prod_intro
tff(fact_7157_open__prod__elim,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B)),Xa: product_prod(A,B)] :
          ( topolo1002775350975398744n_open(product_prod(A,B),S)
         => ( member(product_prod(A,B),Xa,S)
           => ~ ! [A7: set(A)] :
                  ( topolo1002775350975398744n_open(A,A7)
                 => ! [B4: set(B)] :
                      ( topolo1002775350975398744n_open(B,B4)
                     => ( member(product_prod(A,B),Xa,product_Sigma(A,B,A7,aTP_Lamp_agc(set(B),fun(A,set(B)),B4)))
                       => ~ aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A7,aTP_Lamp_agc(set(B),fun(A,set(B)),B4))),S) ) ) ) ) ) ) ).

% open_prod_elim
tff(fact_7158_open__prod__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(product_prod(A,B))] :
          ( topolo1002775350975398744n_open(product_prod(A,B),S)
        <=> ! [X: product_prod(A,B)] :
              ( member(product_prod(A,B),X,S)
             => ? [A8: set(A)] :
                  ( topolo1002775350975398744n_open(A,A8)
                  & ? [B11: set(B)] :
                      ( topolo1002775350975398744n_open(B,B11)
                      & member(product_prod(A,B),X,product_Sigma(A,B,A8,aTP_Lamp_agc(set(B),fun(A,set(B)),B11)))
                      & aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A8,aTP_Lamp_agc(set(B),fun(A,set(B)),B11))),S) ) ) ) ) ) ).

% open_prod_def
tff(fact_7159_principal__prod__principal,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] : prod_filter(A,B,principal(A,A4),principal(B,B2)) = principal(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2))) ).

% principal_prod_principal
tff(fact_7160_Sigma__Image,axiom,
    ! [A: $tType,B: $tType,A4: set(B),B2: fun(B,set(A)),X4: set(B)] : aa(set(B),set(A),image(B,A,product_Sigma(B,A,A4,B2)),X4) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image2(B,set(A),B2),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),X4),A4))) ).

% Sigma_Image
tff(fact_7161_Refl__Field__Restr2,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( aa(set(product_prod(A,A)),set(A),field2(A),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))) = A4 ) ) ) ).

% Refl_Field_Restr2
tff(fact_7162_card__cartesian__product__singleton,axiom,
    ! [A: $tType,B: $tType,Xa: A,A4: set(B)] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),product_Sigma(A,B,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))),aTP_Lamp_agb(set(B),fun(A,set(B)),A4))) = aa(set(B),nat,finite_card(B),A4) ).

% card_cartesian_product_singleton
tff(fact_7163_finite__quotient,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))
       => aa(set(set(A)),$o,finite_finite2(set(A)),equiv_quotient(A,A4,R2)) ) ) ).

% finite_quotient
tff(fact_7164_sum_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(A),B2: fun(A,set(B)),G: fun(A,fun(B,C))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B2,X3)) )
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_agi(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B2),G)),A4) = aa(set(product_prod(A,B)),C,aa(fun(product_prod(A,B),C),fun(set(product_prod(A,B)),C),groups7311177749621191930dd_sum(product_prod(A,B),C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),product_Sigma(A,B,A4,B2)) ) ) ) ) ).

% sum.Sigma
tff(fact_7165_prod_OSigma,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),B2: fun(A,set(B)),G: fun(A,fun(B,C))] :
          ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(set(B),$o,finite_finite2(B),aa(A,set(B),B2,X3)) )
           => ( aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_agj(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),B2),G)),A4) = aa(set(product_prod(A,B)),C,aa(fun(product_prod(A,B),C),fun(set(product_prod(A,B)),C),groups7121269368397514597t_prod(product_prod(A,B),C),aa(fun(A,fun(B,C)),fun(product_prod(A,B),C),product_case_prod(A,B,C),G)),product_Sigma(A,B,A4,B2)) ) ) ) ) ).

% prod.Sigma
tff(fact_7166_relImage__relInvImage,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F2: fun(B,A),A4: set(B)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,aa(set(B),set(A),image2(B,A,F2),A4),aa(set(B),fun(A,set(A)),aTP_Lamp_agk(fun(B,A),fun(set(B),fun(A,set(A))),F2),A4)))
     => ( bNF_Gr4221423524335903396lImage(B,A,bNF_Gr7122648621184425601vImage(B,A,A4,R,F2),F2) = R ) ) ).

% relImage_relInvImage
tff(fact_7167_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_hd(nat,fun(nat,fun(nat,$o)),M))) = product_Sigma(nat,nat,aa(nat,set(nat),set_ord_atMost(nat),M),aTP_Lamp_agl(nat,fun(nat,set(nat)),M)) ).

% pairs_le_eq_Sigma
tff(fact_7168_product__fold,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)) = finite_fold(A,set(product_prod(A,B)),aTP_Lamp_agn(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),B2),bot_bot(set(product_prod(A,B))),A4) ) ) ) ).

% product_fold
tff(fact_7169_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A4: set(A),F2: fun(A,B),B2: set(B)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A4,F2)),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),B2) ) ).

% Gr_incl
tff(fact_7170_init__seg__of__def,axiom,
    ! [A: $tType] : init_seg_of(A) = aa(fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),set(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),collect(product_prod(set(product_prod(A,A)),set(product_prod(A,A)))),aa(fun(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),fun(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),$o),product_case_prod(set(product_prod(A,A)),set(product_prod(A,A)),$o),aTP_Lamp_ago(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)))) ).

% init_seg_of_def
tff(fact_7171_comp__fun__commute__Image__fold,axiom,
    ! [B: $tType,A: $tType,S: set(A)] : finite6289374366891150609ommute(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_agp(set(A),fun(A,fun(B,fun(set(B),set(B)))),S))) ).

% comp_fun_commute_Image_fold
tff(fact_7172_comp__fun__commute__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
     => finite6289374366891150609ommute(product_prod(C,A),set(product_prod(C,B)),aa(fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(C,A),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(C,A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_agr(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),S))) ) ).

% comp_fun_commute_relcomp_fold
tff(fact_7173_Image__fold,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(A)] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => ( aa(set(A),set(B),image(A,B,R),S) = finite_fold(product_prod(A,B),set(B),aa(fun(A,fun(B,fun(set(B),set(B)))),fun(product_prod(A,B),fun(set(B),set(B))),product_case_prod(A,B,fun(set(B),set(B))),aTP_Lamp_agp(set(A),fun(A,fun(B,fun(set(B),set(B)))),S)),bot_bot(set(B)),R) ) ) ).

% Image_fold
tff(fact_7174_lists__length__Suc__eq,axiom,
    ! [A: $tType,A4: set(A),Na: nat] : aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_ags(set(A),fun(nat,fun(list(A),$o)),A4),Na)) = aa(set(product_prod(list(A),A)),set(list(A)),image2(product_prod(list(A),A),list(A),aa(fun(list(A),fun(A,list(A))),fun(product_prod(list(A),A),list(A)),product_case_prod(list(A),A,list(A)),aTP_Lamp_agt(list(A),fun(A,list(A))))),product_Sigma(list(A),A,aa(fun(list(A),$o),set(list(A)),collect(list(A)),aa(nat,fun(list(A),$o),aTP_Lamp_bm(set(A),fun(nat,fun(list(A),$o)),A4),Na)),aTP_Lamp_agu(set(A),fun(list(A),set(A)),A4))) ).

% lists_length_Suc_eq
tff(fact_7175_Restr__natLeq,axiom,
    ! [Na: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Na)),aTP_Lamp_agv(nat,fun(nat,set(nat)),Na))) = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Na))) ).

% Restr_natLeq
tff(fact_7176_wo__rel_Ocases__Total3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A,Phi: fun(A,fun(A,$o))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),R2),id2(A)))
              | member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),A3),aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),R2),id2(A))) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) )
         => ( ( ( A3 = B3 )
             => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) ) ) ) ) ).

% wo_rel.cases_Total3
tff(fact_7177_well__order__induct__imp,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),P: fun(A,$o),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( ! [X3: A] :
            ( ! [Y2: A] :
                ( ( ( Y2 != X3 )
                  & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y2),X3),R2) )
               => ( member(A,Y2,aa(set(product_prod(A,A)),set(A),field2(A),R2))
                 => aa(A,$o,P,Y2) ) )
           => ( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
             => aa(A,$o,P,X3) ) )
       => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => aa(A,$o,P,A3) ) ) ) ).

% well_order_induct_imp
tff(fact_7178_wo__rel_Omax2__among,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => member(A,bNF_We1388413361240627857o_max2(A,R2,A3,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) ) ) ) ).

% wo_rel.max2_among
tff(fact_7179_natLeq__def,axiom,
    bNF_Ca8665028551170535155natLeq = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),ord_less_eq(nat))) ).

% natLeq_def
tff(fact_7180_wo__rel_Ocases__Total,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A,Phi: fun(A,fun(A,$o))] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2)
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) )
         => ( ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),A3),R2)
             => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) )
           => aa(A,$o,aa(A,fun(A,$o),Phi,A3),B3) ) ) ) ) ).

% wo_rel.cases_Total
tff(fact_7181_natLeq__on__wo__rel,axiom,
    ! [Na: nat] : bNF_Wellorder_wo_rel(nat,aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Na)))) ).

% natLeq_on_wo_rel
tff(fact_7182_wo__rel_Omax2__greater__among,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),bNF_We1388413361240627857o_max2(A,R2,A3,B3)),R2)
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),bNF_We1388413361240627857o_max2(A,R2,A3,B3)),R2)
            & member(A,bNF_We1388413361240627857o_max2(A,R2,A3,B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) ) ) ) ) ).

% wo_rel.max2_greater_among
tff(fact_7183_Restr__natLeq2,axiom,
    ! [Na: nat] : aa(set(product_prod(nat,nat)),set(product_prod(nat,nat)),aa(set(product_prod(nat,nat)),fun(set(product_prod(nat,nat)),set(product_prod(nat,nat))),inf_inf(set(product_prod(nat,nat))),bNF_Ca8665028551170535155natLeq),product_Sigma(nat,nat,order_underS(nat,bNF_Ca8665028551170535155natLeq,Na),aTP_Lamp_agw(nat,fun(nat,set(nat)),Na))) = aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Na))) ).

% Restr_natLeq2
tff(fact_7184_wo__rel_OWell__order__isMinim__exists,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( B2 != bot_bot(set(A)) )
         => ? [X_1: A] : bNF_We4791949203932849705sMinim(A,R2,B2,X_1) ) ) ) ).

% wo_rel.Well_order_isMinim_exists
tff(fact_7185_natLeq__underS__less,axiom,
    ! [Na: nat] : order_underS(nat,bNF_Ca8665028551170535155natLeq,Na) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Na)) ).

% natLeq_underS_less
tff(fact_7186_underS__Field2,axiom,
    ! [A: $tType,A3: A,R2: set(product_prod(A,A))] :
      ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).

% underS_Field2
tff(fact_7187_BNF__Least__Fixpoint_OunderS__Field,axiom,
    ! [A: $tType,I: A,R: set(product_prod(A,A)),J: A] :
      ( member(A,I,order_underS(A,R,J))
     => member(A,I,aa(set(product_prod(A,A)),set(A),field2(A),R)) ) ).

% BNF_Least_Fixpoint.underS_Field
tff(fact_7188_underS__E,axiom,
    ! [A: $tType,I: A,R: set(product_prod(A,A)),J: A] :
      ( member(A,I,order_underS(A,R,J))
     => ( ( I != J )
        & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,I),J),R) ) ) ).

% underS_E
tff(fact_7189_underS__I,axiom,
    ! [A: $tType,I: A,J: A,R: set(product_prod(A,A))] :
      ( ( I != J )
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,I),J),R)
       => member(A,I,order_underS(A,R,J)) ) ) ).

% underS_I
tff(fact_7190_underS__empty,axiom,
    ! [A: $tType,A3: A,R2: set(product_prod(A,A))] :
      ( ~ member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( order_underS(A,R2,A3) = bot_bot(set(A)) ) ) ).

% underS_empty
tff(fact_7191_Order__Relation_OunderS__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ).

% Order_Relation.underS_Field
tff(fact_7192_underS__Field3,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),order_underS(A,R2,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).

% underS_Field3
tff(fact_7193_underS__incl__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_underS(A,R2,B3))
          <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2) ) ) ) ) ).

% underS_incl_iff
tff(fact_7194_wo__rel_Ominim__isMinim,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( B2 != bot_bot(set(A)) )
         => bNF_We4791949203932849705sMinim(A,R2,B2,bNF_We6954850376910717587_minim(A,R2,B2)) ) ) ) ).

% wo_rel.minim_isMinim
tff(fact_7195_Refl__under__underS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( refl_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( order_under(A,R2,A3) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),order_underS(A,R2,A3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))) ) ) ) ).

% Refl_under_underS
tff(fact_7196_underS__subset__under,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_under(A,R2,A3)) ).

% underS_subset_under
tff(fact_7197_under__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_under(A,R2,A3)),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ).

% under_Field
tff(fact_7198_wo__rel_Ominim__least,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A),B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,B3,B2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,bNF_We6954850376910717587_minim(A,R2,B2)),B3),R2) ) ) ) ).

% wo_rel.minim_least
tff(fact_7199_wo__rel_Oequals__minim,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,A3,B2)
         => ( ! [B5: A] :
                ( member(A,B5,B2)
               => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B5),R2) )
           => ( A3 = bNF_We6954850376910717587_minim(A,R2,B2) ) ) ) ) ) ).

% wo_rel.equals_minim
tff(fact_7200_wo__rel_Ominim__inField,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( B2 != bot_bot(set(A)) )
         => member(A,bNF_We6954850376910717587_minim(A,R2,B2),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ) ) ).

% wo_rel.minim_inField
tff(fact_7201_wo__rel_Ominim__in,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( B2 != bot_bot(set(A)) )
         => member(A,bNF_We6954850376910717587_minim(A,R2,B2),B2) ) ) ) ).

% wo_rel.minim_in
tff(fact_7202_relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,B)),S: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R)
     => ( aa(set(product_prod(B,C)),$o,finite_finite2(product_prod(B,C)),S)
       => ( relcomp(A,B,C,R,S) = finite_fold(product_prod(A,B),set(product_prod(A,C)),aa(fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(A,B),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(A,B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_agy(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),S)),bot_bot(set(product_prod(A,C))),R) ) ) ) ).

% relcomp_fold
tff(fact_7203_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_7204_relcomp__empty2,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(A,C))] : relcomp(A,C,B,R,bot_bot(set(product_prod(C,B)))) = bot_bot(set(product_prod(A,B))) ).

% relcomp_empty2
tff(fact_7205_relcomp__empty1,axiom,
    ! [C: $tType,B: $tType,A: $tType,R: set(product_prod(C,B))] : relcomp(A,C,B,bot_bot(set(product_prod(A,C))),R) = bot_bot(set(product_prod(A,B))) ).

% relcomp_empty1
tff(fact_7206_relcomp__distrib,axiom,
    ! [A: $tType,B: $tType,C: $tType,R: set(product_prod(A,C)),S: set(product_prod(C,B)),T3: set(product_prod(C,B))] : relcomp(A,C,B,R,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),S),T3)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),relcomp(A,C,B,R,S)),relcomp(A,C,B,R,T3)) ).

% relcomp_distrib
tff(fact_7207_relcomp__distrib2,axiom,
    ! [A: $tType,B: $tType,C: $tType,S: set(product_prod(A,C)),T3: set(product_prod(A,C)),R: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),set(product_prod(A,C))),sup_sup(set(product_prod(A,C))),S),T3),R) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),relcomp(A,C,B,S,R)),relcomp(A,C,B,T3,R)) ).

% relcomp_distrib2
tff(fact_7208_R__O__Id,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : relcomp(A,B,B,R,id2(B)) = R ).

% R_O_Id
tff(fact_7209_Id__O__R,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : relcomp(A,A,B,id2(A),R) = R ).

% Id_O_R
tff(fact_7210_range__snd,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(A)) ).

% range_snd
tff(fact_7211_snd__image__times,axiom,
    ! [B: $tType,A: $tType,A4: set(B),B2: set(A)] :
      aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),product_Sigma(B,A,A4,aTP_Lamp_ld(set(A),fun(B,set(A)),B2))) = $ite(A4 = bot_bot(set(B)),bot_bot(set(A)),B2) ).

% snd_image_times
tff(fact_7212_relcomp__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,B)),A4: set(A),B2: set(B),S3: set(product_prod(B,C)),C2: set(C)] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))
     => ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),S3),product_Sigma(B,C,B2,aTP_Lamp_agz(set(C),fun(B,set(C)),C2)))
       => aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R2,S3)),product_Sigma(A,C,A4,aTP_Lamp_aha(set(C),fun(A,set(C)),C2))) ) ) ).

% relcomp_subset_Sigma
tff(fact_7213_relcomp__UNION__distrib,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,S3: set(product_prod(A,C)),R2: fun(D,set(product_prod(C,B))),I5: set(D)] : relcomp(A,C,B,S3,aa(set(set(product_prod(C,B))),set(product_prod(C,B)),complete_Sup_Sup(set(product_prod(C,B))),aa(set(D),set(set(product_prod(C,B))),image2(D,set(product_prod(C,B)),R2),I5))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_ahb(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),S3),R2)),I5)) ).

% relcomp_UNION_distrib
tff(fact_7214_relcomp__UNION__distrib2,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType,R2: fun(D,set(product_prod(A,C))),I5: set(D),S3: set(product_prod(C,B))] : relcomp(A,C,B,aa(set(set(product_prod(A,C))),set(product_prod(A,C)),complete_Sup_Sup(set(product_prod(A,C))),aa(set(D),set(set(product_prod(A,C))),image2(D,set(product_prod(A,C)),R2),I5)),S3) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(D),set(set(product_prod(A,B))),image2(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_ahc(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),R2),S3)),I5)) ).

% relcomp_UNION_distrib2
tff(fact_7215_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_7216_relcomp_Ocases,axiom,
    ! [A: $tType,B: $tType,C: $tType,A12: A,A23: B,R2: set(product_prod(A,C)),S3: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A12),A23),relcomp(A,C,B,R2,S3))
     => ~ ! [B5: C] :
            ( member(product_prod(A,C),aa(C,product_prod(A,C),product_Pair(A,C,A12),B5),R2)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),product_Pair(C,B,B5),A23),S3) ) ) ).

% relcomp.cases
tff(fact_7217_relcomp_Osimps,axiom,
    ! [A: $tType,B: $tType,C: $tType,A12: A,A23: B,R2: set(product_prod(A,C)),S3: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A12),A23),relcomp(A,C,B,R2,S3))
    <=> ? [A9: A,B7: C,C6: B] :
          ( ( A12 = A9 )
          & ( A23 = C6 )
          & member(product_prod(A,C),aa(C,product_prod(A,C),product_Pair(A,C,A9),B7),R2)
          & member(product_prod(C,B),aa(B,product_prod(C,B),product_Pair(C,B,B7),C6),S3) ) ) ).

% relcomp.simps
tff(fact_7218_relcomp_OrelcompI,axiom,
    ! [A: $tType,C: $tType,B: $tType,A3: A,B3: B,R2: set(product_prod(A,B)),C3: C,S3: set(product_prod(B,C))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),R2)
     => ( member(product_prod(B,C),aa(C,product_prod(B,C),product_Pair(B,C,B3),C3),S3)
       => member(product_prod(A,C),aa(C,product_prod(A,C),product_Pair(A,C,A3),C3),relcomp(A,B,C,R2,S3)) ) ) ).

% relcomp.relcompI
tff(fact_7219_relcompE,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xz: product_prod(A,B),R2: set(product_prod(A,C)),S3: set(product_prod(C,B))] :
      ( member(product_prod(A,B),Xz,relcomp(A,C,B,R2,S3))
     => ~ ! [X3: A,Y: C,Z2: B] :
            ( ( Xz = aa(B,product_prod(A,B),product_Pair(A,B,X3),Z2) )
           => ( member(product_prod(A,C),aa(C,product_prod(A,C),product_Pair(A,C,X3),Y),R2)
             => ~ member(product_prod(C,B),aa(B,product_prod(C,B),product_Pair(C,B,Y),Z2),S3) ) ) ) ).

% relcompE
tff(fact_7220_relcompEpair,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: A,C3: B,R2: set(product_prod(A,C)),S3: set(product_prod(C,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),C3),relcomp(A,C,B,R2,S3))
     => ~ ! [B5: C] :
            ( member(product_prod(A,C),aa(C,product_prod(A,C),product_Pair(A,C,A3),B5),R2)
           => ~ member(product_prod(C,B),aa(B,product_prod(C,B),product_Pair(C,B,B5),C3),S3) ) ) ).

% relcompEpair
tff(fact_7221_O__assoc,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,R: set(product_prod(A,D)),S: set(product_prod(D,C)),T3: set(product_prod(C,B))] : relcomp(A,C,B,relcomp(A,D,C,R,S),T3) = relcomp(A,D,B,R,relcomp(D,C,B,S,T3)) ).

% O_assoc
tff(fact_7222_relcomp__Image,axiom,
    ! [A: $tType,C: $tType,B: $tType,X4: set(product_prod(B,C)),Y3: set(product_prod(C,A)),Z6: set(B)] : aa(set(B),set(A),image(B,A,relcomp(B,C,A,X4,Y3)),Z6) = aa(set(C),set(A),image(C,A,Y3),aa(set(B),set(C),image(B,C,X4),Z6)) ).

% relcomp_Image
tff(fact_7223_min__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert2(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R)) ) ).

% min_ext_compat
tff(fact_7224_max__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
     => aa(set(product_prod(set(A),set(A))),$o,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),$o),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,S)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert2(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R)) ) ).

% max_ext_compat
tff(fact_7225_relcomp__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType,R4: set(product_prod(A,B)),R2: set(product_prod(A,B)),S5: set(product_prod(B,C)),S3: set(product_prod(B,C))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R4),R2)
     => ( aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),S5),S3)
       => aa(set(product_prod(A,C)),$o,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),$o),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R4,S5)),relcomp(A,B,C,R2,S3)) ) ) ).

% relcomp_mono
tff(fact_7226_wf__relcomp__compatible,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),relcomp(A,A,A,S,R))
       => wf(A,relcomp(A,A,A,S,R)) ) ) ).

% wf_relcomp_compatible
tff(fact_7227_wf__union__compatible,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( wf(A,S)
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
         => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),S)) ) ) ) ).

% wf_union_compatible
tff(fact_7228_trancl__Int__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S3)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_trancl(A,R2)),S3),R2)),S3)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R2)),S3) ) ) ).

% trancl_Int_subset
tff(fact_7229_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R2: set(product_prod(A,C)),S3: set(product_prod(C,B))] : relcomp(A,C,B,R2,S3) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_ahd(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),R2),S3))) ).

% relcomp_unfold
tff(fact_7230_qc__wf__relto__iff,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),relcomp(A,A,A,transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),S)),R))
     => ( wf(A,relcomp(A,A,A,transitive_rtrancl(A,S),relcomp(A,A,A,R,transitive_rtrancl(A,S))))
      <=> wf(A,R) ) ) ).

% qc_wf_relto_iff
tff(fact_7231_rtrancl__Int__subset,axiom,
    ! [A: $tType,S3: set(product_prod(A,A)),R2: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),id2(A)),S3)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_rtrancl(A,R2)),S3),R2)),S3)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R2)),S3) ) ) ).

% rtrancl_Int_subset
tff(fact_7232_subset__snd__imageI,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B),S: set(product_prod(A,B)),Xa: A] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2))),S)
     => ( member(A,Xa,A4)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),S)) ) ) ).

% subset_snd_imageI
tff(fact_7233_reduction__pairI,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( wf(A,R)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S)),R)
       => fun_reduction_pair(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)),R),S)) ) ) ).

% reduction_pairI
tff(fact_7234_insert__relcomp__union__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),Xa: product_prod(C,A),X4: set(product_prod(C,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
     => ( aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(set(product_prod(C,B)),fun(set(product_prod(C,B)),set(product_prod(C,B))),sup_sup(set(product_prod(C,B))),relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert2(product_prod(C,A)),Xa),bot_bot(set(product_prod(C,A)))),S)),X4) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ahe(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xa)),X4,S) ) ) ).

% insert_relcomp_union_fold
tff(fact_7235_range__fst,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),top_top(set(product_prod(A,B)))) = top_top(set(A)) ).

% range_fst
tff(fact_7236_fst__image__times,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2))) = $ite(B2 = bot_bot(set(B)),bot_bot(set(A)),A4) ).

% fst_image_times
tff(fact_7237_Suc__0__div__numeral,axiom,
    ! [K: num] : 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_7238_Id__fstsnd__eq,axiom,
    ! [A: $tType] : id2(A) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aTP_Lamp_ahf(product_prod(A,A),$o)) ).

% Id_fstsnd_eq
tff(fact_7239_reduction__pair__def,axiom,
    ! [A: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A)))] :
      ( fun_reduction_pair(A,P)
    <=> ( wf(A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P))
        & aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P))),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P)) ) ) ).

% reduction_pair_def
tff(fact_7240_reduction__pair__lemma,axiom,
    ! [A: $tType,P: product_prod(set(product_prod(A,A)),set(product_prod(A,A))),R: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( fun_reduction_pair(A,P)
     => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_fst(set(product_prod(A,A)),set(product_prod(A,A))),P))
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),S),aa(product_prod(set(product_prod(A,A)),set(product_prod(A,A))),set(product_prod(A,A)),product_snd(set(product_prod(A,A)),set(product_prod(A,A))),P))
         => ( wf(A,S)
           => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),S)) ) ) ) ) ).

% reduction_pair_lemma
tff(fact_7241_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_7242_fst__image__Sigma,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: fun(A,set(B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),product_Sigma(A,B,A4,B2)) = aa(fun(A,$o),set(A),collect(A),aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_agg(set(A),fun(fun(A,set(B)),fun(A,$o)),A4),B2)) ).

% fst_image_Sigma
tff(fact_7243_sorted__enumerate,axiom,
    ! [A: $tType,Na: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),map(product_prod(nat,A),nat,product_fst(nat,A),enumerate(A,Na,Xs))) ).

% sorted_enumerate
tff(fact_7244_rel__fun__Collect__case__prodD,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,A4: fun(A,fun(B,$o)),B2: fun(C,fun(D,$o)),F2: fun(A,C),G: fun(B,D),X4: set(product_prod(A,B)),Xa: product_prod(A,B)] :
      ( aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,A4,B2),F2),G)
     => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),X4),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),A4)))
       => ( member(product_prod(A,B),Xa,X4)
         => aa(D,$o,aa(C,fun(D,$o),B2,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)),Xa)),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)),Xa)) ) ) ) ).

% rel_fun_Collect_case_prodD
tff(fact_7245_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X4: set(A),A4: set(product_prod(A,B)),Y3: set(B),P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ( X4 = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A4) )
     => ( ( Y3 = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),A4) )
       => ( ! [X3: A] :
              ( member(A,X3,X4)
             => ! [Xa4: B] :
                  ( member(B,Xa4,Y3)
                 => ( aa(B,$o,aa(A,fun(B,$o),P,X3),Xa4)
                   => aa(B,$o,aa(A,fun(B,$o),Q,X3),Xa4) ) ) )
         => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A4),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P)))
           => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A4),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Q))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_7246_subset__fst__imageI,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B),S: set(product_prod(A,B)),Ya: B] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2))),S)
     => ( member(B,Ya,B2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),S)) ) ) ).

% subset_fst_imageI
tff(fact_7247_bezw__non__0,axiom,
    ! [Ya: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Ya)
     => ( bezw(Xa,Ya) = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Ya,modulo_modulo(nat,Xa,Ya)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Ya,modulo_modulo(nat,Xa,Ya)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Ya,modulo_modulo(nat,Xa,Ya)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xa,Ya))))) ) ) ).

% bezw_non_0
tff(fact_7248_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B))] : aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A4),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),A4),aTP_Lamp_ahg(set(product_prod(A,B)),fun(A,set(B)),A4))) ).

% subset_fst_snd
tff(fact_7249_eventually__prodI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F3: filter(A),Q: fun(B,$o),G2: filter(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Q),G2)
       => aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_ahh(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),P),Q)),prod_filter(A,B,F3,G2)) ) ) ).

% eventually_prodI
tff(fact_7250_bezw_Osimps,axiom,
    ! [Xa: nat,Ya: nat] :
      bezw(Xa,Ya) = $ite(Ya = zero_zero(nat),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Ya,modulo_modulo(nat,Xa,Ya)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Ya,modulo_modulo(nat,Xa,Ya)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Ya,modulo_modulo(nat,Xa,Ya)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xa,Ya)))))) ).

% bezw.simps
tff(fact_7251_bezw_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: product_prod(int,int)] :
      ( ( bezw(Xa,Xaa) = Ya )
     => ( Ya = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xa,Xaa)))))) ) ) ).

% bezw.elims
tff(fact_7252_predicate2__transferD,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R1: fun(A,fun(B,$o)),R22: fun(C,fun(D,$o)),P: fun(A,fun(C,$o)),Q: fun(B,fun(D,$o)),A3: product_prod(A,B),A4: set(product_prod(A,B)),B3: product_prod(C,D),B2: set(product_prod(C,D))] :
      ( aa(fun(B,fun(D,$o)),$o,aa(fun(A,fun(C,$o)),fun(fun(B,fun(D,$o)),$o),bNF_rel_fun(A,B,fun(C,$o),fun(D,$o),R1,bNF_rel_fun(C,D,$o,$o,R22,fequal($o))),P),Q)
     => ( member(product_prod(A,B),A3,A4)
       => ( member(product_prod(C,D),B3,B2)
         => ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A4),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R1)))
           => ( aa(set(product_prod(C,D)),$o,aa(set(product_prod(C,D)),fun(set(product_prod(C,D)),$o),ord_less_eq(set(product_prod(C,D))),B2),aa(fun(product_prod(C,D),$o),set(product_prod(C,D)),collect(product_prod(C,D)),aa(fun(C,fun(D,$o)),fun(product_prod(C,D),$o),product_case_prod(C,D,$o),R22)))
             => ( aa(C,$o,aa(A,fun(C,$o),P,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(C,D),C,product_fst(C,D),B3))
              <=> aa(D,$o,aa(B,fun(D,$o),Q,aa(product_prod(A,B),B,product_snd(A,B),A3)),aa(product_prod(C,D),D,product_snd(C,D),B3)) ) ) ) ) ) ) ).

% predicate2_transferD
tff(fact_7253_nths__shift__lemma,axiom,
    ! [A: $tType,A4: set(nat),Xs: list(A),I: nat] : map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_ahi(set(nat),fun(product_prod(A,nat),$o),A4),zip(A,nat,Xs,upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(list(A),nat,size_size(list(A)),Xs)))))) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_ahj(set(nat),fun(nat,fun(product_prod(A,nat),$o)),A4),I),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_7254_nths__def,axiom,
    ! [A: $tType,Xs: list(A),A4: set(nat)] : nths(A,Xs,A4) = map(product_prod(A,nat),A,product_fst(A,nat),filter2(product_prod(A,nat),aTP_Lamp_ahi(set(nat),fun(product_prod(A,nat),$o),A4),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_7255_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P3: product_prod(A,B),Xs: list(A),Ys2: list(B)] :
      ( member(product_prod(A,B),P3,aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys2)))
    <=> ? [N: nat] :
          ( ( aa(nat,A,nth(A,Xs),N) = aa(product_prod(A,B),A,product_fst(A,B),P3) )
          & ( aa(nat,B,nth(B,Ys2),N) = aa(product_prod(A,B),B,product_snd(A,B),P3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs))
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N),aa(list(B),nat,size_size(list(B)),Ys2)) ) ) ).

% in_set_zip
tff(fact_7256_fun_Oin__rel,axiom,
    ! [B: $tType,C: $tType,A: $tType,R: fun(B,fun(C,$o)),A3: fun(A,B),B3: fun(A,C)] :
      ( aa(fun(A,C),$o,aa(fun(A,B),fun(fun(A,C),$o),bNF_rel_fun(A,A,B,C,fequal(A),R),A3),B3)
    <=> ? [Z5: fun(A,product_prod(B,C))] :
          ( member(fun(A,product_prod(B,C)),Z5,aa(fun(fun(A,product_prod(B,C)),$o),set(fun(A,product_prod(B,C))),collect(fun(A,product_prod(B,C))),aTP_Lamp_ahk(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),R)))
          & ( aa(fun(A,product_prod(B,C)),fun(A,B),comp(product_prod(B,C),B,A,product_fst(B,C)),Z5) = A3 )
          & ( aa(fun(A,product_prod(B,C)),fun(A,C),comp(product_prod(B,C),C,A,product_snd(B,C)),Z5) = B3 ) ) ) ).

% fun.in_rel
tff(fact_7257_insert__relcomp__fold,axiom,
    ! [C: $tType,B: $tType,A: $tType,S: set(product_prod(A,B)),Xa: product_prod(C,A),R: set(product_prod(C,A))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),S)
     => ( relcomp(C,A,B,aa(set(product_prod(C,A)),set(product_prod(C,A)),aa(product_prod(C,A),fun(set(product_prod(C,A)),set(product_prod(C,A))),insert2(product_prod(C,A)),Xa),R),S) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ahe(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Xa)),relcomp(C,A,B,R,S),S) ) ) ).

% insert_relcomp_fold
tff(fact_7258_in__set__enumerate__eq,axiom,
    ! [A: $tType,P3: product_prod(nat,A),Na: nat,Xs: list(A)] :
      ( member(product_prod(nat,A),P3,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Na,Xs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),aa(product_prod(nat,A),nat,product_fst(nat,A),P3))
        & aa(nat,$o,aa(nat,fun(nat,$o),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)),Na))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(product_prod(nat,A),nat,product_fst(nat,A),P3)),Na)) = aa(product_prod(nat,A),A,product_snd(nat,A),P3) ) ) ) ).

% in_set_enumerate_eq
tff(fact_7259_bezw_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: product_prod(int,int)] :
      ( ( bezw(Xa,Xaa) = Ya )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Xaa))
       => ~ ( ( Ya = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,modulo_modulo(nat,Xa,Xaa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,Xa,Xaa)))))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_7260_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_7261_Rat_Opositive_Orep__eq,axiom,
    ! [Xa: rat] :
      ( aa(rat,$o,positive,Xa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,Xa))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,Xa)))) ) ).

% Rat.positive.rep_eq
tff(fact_7262_normalize__def,axiom,
    ! [P3: product_prod(int,int)] :
      normalize(P3) = $ite(
        aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P3)),
        $let(
          a3: int,
          a3:= 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,product_prod(int,int),product_Pair(int,int,divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P3),a3)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P3),a3)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P3) = zero_zero(int),
          aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)),
          $let(
            a3: int,
            a3:= 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,product_prod(int,int),product_Pair(int,int,divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P3),a3)),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P3),a3)) ) ) ) ).

% normalize_def
tff(fact_7263_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B3 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_7264_gcd__pos__int,axiom,
    ! [M: int,Na: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),Na))
    <=> ( ( M != zero_zero(int) )
        | ( Na != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_7265_Gcd__insert,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,A4: set(A)] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),gcd_Gcd(A,A4)) ) ).

% Gcd_insert
tff(fact_7266_Gcd__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4)) ) ).

% Gcd_fin.insert
tff(fact_7267_Gcd__2,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: A,B3: A] : gcd_Gcd(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) ) ).

% Gcd_2
tff(fact_7268_Gcd__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [B2: set(A),A4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
         => ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),B2)),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4)) = aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) ) ) ) ).

% Gcd_fin.subset
tff(fact_7269_gcd__ge__0__int,axiom,
    ! [Xa: int,Ya: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),Ya)) ).

% gcd_ge_0_int
tff(fact_7270_Gcd__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A),B2: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(set(A),A,semiring_gcd_Gcd_fin(A),A4)),aa(set(A),A,semiring_gcd_Gcd_fin(A),B2)) ) ).

% Gcd_fin.union
tff(fact_7271_gcd__le1__int,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),A3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B3)),A3) ) ).

% gcd_le1_int
tff(fact_7272_gcd__le2__int,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B3)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B3)),B3) ) ).

% gcd_le2_int
tff(fact_7273_gcd__cases__int,axiom,
    ! [Xa: int,Ya: int,P: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
         => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),Ya)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xa)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ya),zero_zero(int))
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),aa(int,int,uminus_uminus(int),Ya))) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Ya)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),Xa)),Ya)) ) )
         => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xa),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ya),zero_zero(int))
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),Xa)),aa(int,int,uminus_uminus(int),Ya))) ) )
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),Ya)) ) ) ) ) ).

% gcd_cases_int
tff(fact_7274_gcd__unique__int,axiom,
    ! [D2: int,A3: int,B3: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),A3)
        & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),D2),B3)
        & ! [E3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E3),A3)
              & aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E3),B3) )
           => aa(int,$o,aa(int,fun(int,$o),dvd_dvd(int),E3),D2) ) )
    <=> ( D2 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A3),B3) ) ) ).

% gcd_unique_int
tff(fact_7275_gcd__non__0__int,axiom,
    ! [Ya: int,Xa: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Ya)
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Xa),Ya) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Ya),modulo_modulo(int,Xa,Ya)) ) ) ).

% gcd_non_0_int
tff(fact_7276_Gcd__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ).

% Gcd_fin.insert_remove
tff(fact_7277_Gcd__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,A4: set(A)] :
          ( member(A,A3,A4)
         => ( aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% Gcd_fin.remove
tff(fact_7278_Gcd__set__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [Xs: list(A)] : gcd_Gcd(A,aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_gcd(A),Xs,zero_zero(A)) ) ).

% Gcd_set_eq_fold
tff(fact_7279_Gcd__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Xs: list(A)] : aa(set(A),A,semiring_gcd_Gcd_fin(A),aa(list(A),set(A),set2(A),Xs)) = fold(A,A,gcd_gcd(A),Xs,zero_zero(A)) ) ).

% Gcd_fin.set_eq_fold
tff(fact_7280_Gcd__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A4: set(A)] :
          aa(set(A),A,semiring_gcd_Gcd_fin(A),A4) = $ite(aa(set(A),$o,finite_finite2(A),A4),finite_fold(A,A,gcd_gcd(A),zero_zero(A),A4),one_one(A)) ) ).

% Gcd_fin.eq_fold
tff(fact_7281_Rat_Opositive_Otransfer,axiom,
    aa(fun(rat,$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(rat,$o),$o),bNF_rel_fun(product_prod(int,int),rat,$o,$o,pcr_rat,fequal($o)),aTP_Lamp_ahl(product_prod(int,int),$o)),positive) ).

% Rat.positive.transfer
tff(fact_7282_quotient__of__def,axiom,
    ! [Xa: rat] : quotient_of(Xa) = the(product_prod(int,int),aTP_Lamp_ahm(rat,fun(product_prod(int,int),$o),Xa)) ).

% quotient_of_def
tff(fact_7283_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B3) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_7284_gcd__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),A3) = A3 ).

% gcd_nat.left_neutral
tff(fact_7285_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B3) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_7286_gcd__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),zero_zero(nat)) = A3 ).

% gcd_nat.right_neutral
tff(fact_7287_gcd__0__nat,axiom,
    ! [Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),zero_zero(nat)) = Xa ).

% gcd_0_nat
tff(fact_7288_gcd__0__left__nat,axiom,
    ! [Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),zero_zero(nat)),Xa) = Xa ).

% gcd_0_left_nat
tff(fact_7289_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_7290_gcd__pos__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),Na))
    <=> ( ( M != zero_zero(nat) )
        | ( Na != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_7291_coprime__mod__right__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,A3,modulo_modulo(A,B3,A3))
          <=> algebr8660921524188924756oprime(A,A3,B3) ) ) ) ).

% coprime_mod_right_iff
tff(fact_7292_coprime__mod__left__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( algebr8660921524188924756oprime(A,modulo_modulo(A,A3,B3),B3)
          <=> algebr8660921524188924756oprime(A,A3,B3) ) ) ) ).

% coprime_mod_left_iff
tff(fact_7293_coprime__power__left__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,Na: nat,B3: A] :
          ( algebr8660921524188924756oprime(A,aa(nat,A,power_power(A,A3),Na),B3)
        <=> ( algebr8660921524188924756oprime(A,A3,B3)
            | ( Na = zero_zero(nat) ) ) ) ) ).

% coprime_power_left_iff
tff(fact_7294_coprime__power__right__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,Na: nat] :
          ( algebr8660921524188924756oprime(A,A3,aa(nat,A,power_power(A,B3),Na))
        <=> ( algebr8660921524188924756oprime(A,A3,B3)
            | ( Na = zero_zero(nat) ) ) ) ) ).

% coprime_power_right_iff
tff(fact_7295_coprime__0__left__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,zero_zero(A),A3)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A)) ) ) ).

% coprime_0_left_iff
tff(fact_7296_coprime__0__right__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( algebr8660921524188924756oprime(A,A3,zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A)) ) ) ).

% coprime_0_right_iff
tff(fact_7297_normalize__stable,axiom,
    ! [Q5: int,P3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),Q5)
     => ( algebr8660921524188924756oprime(int,P3,Q5)
       => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P3),Q5)) = aa(int,product_prod(int,int),product_Pair(int,int,P3),Q5) ) ) ) ).

% normalize_stable
tff(fact_7298_gcd__diff2__nat,axiom,
    ! [M: nat,Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,minus_minus(nat,Na),M)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),Na) ) ) ).

% gcd_diff2_nat
tff(fact_7299_gcd__diff1__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Na),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,minus_minus(nat,M),Na)),Na) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),Na) ) ) ).

% gcd_diff1_nat
tff(fact_7300_gcd__le1__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B3)),A3) ) ).

% gcd_le1_nat
tff(fact_7301_gcd__le2__nat,axiom,
    ! [B3: nat,A3: nat] :
      ( ( B3 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B3)),B3) ) ).

% gcd_le2_nat
tff(fact_7302_gcd__non__0__nat,axiom,
    ! [Ya: nat,Xa: nat] :
      ( ( Ya != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Ya) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ya),modulo_modulo(nat,Xa,Ya)) ) ) ).

% gcd_non_0_nat
tff(fact_7303_gcd__nat_Osimps,axiom,
    ! [Xa: nat,Ya: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Ya) = $ite(Ya = zero_zero(nat),Xa,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Ya),modulo_modulo(nat,Xa,Ya))) ).

% gcd_nat.simps
tff(fact_7304_gcd__nat_Oelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Xaa) = Ya )
     => ( Ya = $ite(Xaa = zero_zero(nat),Xa,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xa,Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_7305_div__gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] :
          ( ( ( A3 != zero_zero(A) )
            | ( B3 != zero_zero(A) ) )
         => algebr8660921524188924756oprime(A,divide_divide(A,A3,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)),divide_divide(A,B3,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3))) ) ) ).

% div_gcd_coprime
tff(fact_7306_gcd__coprime__exists,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) != zero_zero(A) )
         => ? [A20: A,B14: A] :
              ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A20),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) )
              & ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),B14),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) )
              & algebr8660921524188924756oprime(A,A20,B14) ) ) ) ).

% gcd_coprime_exists
tff(fact_7307_gcd__coprime,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,A11: A,B9: A] :
          ( ( aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3) != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),A11),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) )
           => ( ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),B9),aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A3),B3)) )
             => algebr8660921524188924756oprime(A,A11,B9) ) ) ) ) ).

% gcd_coprime
tff(fact_7308_bezout__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [X3: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B3)) ) ).

% bezout_nat
tff(fact_7309_Gcd__in,axiom,
    ! [A4: set(nat)] :
      ( ! [A5: nat,B5: nat] :
          ( member(nat,A5,A4)
         => ( member(nat,B5,A4)
           => member(nat,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A5),B5),A4) ) )
     => ( ( A4 != bot_bot(set(nat)) )
       => member(nat,gcd_Gcd(nat,A4),A4) ) ) ).

% Gcd_in
tff(fact_7310_Rat__cases,axiom,
    ! [Q5: rat] :
      ~ ! [A5: int,B5: int] :
          ( ( Q5 = fract(A5,B5) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
           => ~ algebr8660921524188924756oprime(int,A5,B5) ) ) ).

% Rat_cases
tff(fact_7311_Rat__induct,axiom,
    ! [P: fun(rat,$o),Q5: rat] :
      ( ! [A5: int,B5: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
         => ( algebr8660921524188924756oprime(int,A5,B5)
           => aa(rat,$o,P,fract(A5,B5)) ) )
     => aa(rat,$o,P,Q5) ) ).

% Rat_induct
tff(fact_7312_bezout__gcd__nat_H,axiom,
    ! [B3: nat,A3: nat] :
    ? [X3: nat,Y: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B3) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X3))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A3),B3) ) ) ) ).

% bezout_gcd_nat'
tff(fact_7313_Gcd__nat__set__eq__fold,axiom,
    ! [Xs: list(nat)] : gcd_Gcd(nat,aa(list(nat),set(nat),set2(nat),Xs)) = fold(nat,nat,gcd_gcd(nat),Xs,zero_zero(nat)) ).

% Gcd_nat_set_eq_fold
tff(fact_7314_Rat__cases__nonzero,axiom,
    ! [Q5: rat] :
      ( ! [A5: int,B5: int] :
          ( ( Q5 = fract(A5,B5) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
           => ( ( A5 != zero_zero(int) )
             => ~ algebr8660921524188924756oprime(int,A5,B5) ) ) )
     => ( Q5 = zero_zero(rat) ) ) ).

% Rat_cases_nonzero
tff(fact_7315_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,gcd_gcd(nat),zero_zero(nat),dvd_dvd(nat),aTP_Lamp_ahn(nat,fun(nat,$o))) ).

% gcd_nat.semilattice_neutr_order_axioms
tff(fact_7316_gcd__is__Max__divisors__nat,axiom,
    ! [Na: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),Na) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_aho(nat,fun(nat,fun(nat,$o)),Na),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_7317_Rats__cases_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Xa: A] :
          ( member(A,Xa,field_char_0_Rats(A))
         => ~ ! [A5: int,B5: int] :
                ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),B5)
               => ( algebr8660921524188924756oprime(int,A5,B5)
                 => ( Xa != divide_divide(A,aa(int,A,ring_1_of_int(A),A5),aa(int,A,ring_1_of_int(A),B5)) ) ) ) ) ) ).

% Rats_cases'
tff(fact_7318_quotient__of__unique,axiom,
    ! [R2: rat] :
    ? [X3: product_prod(int,int)] :
      ( ( R2 = fract(aa(product_prod(int,int),int,product_fst(int,int),X3),aa(product_prod(int,int),int,product_snd(int,int),X3)) )
      & aa(int,$o,aa(int,fun(int,$o),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))
      & ! [Y2: product_prod(int,int)] :
          ( ( ( R2 = fract(aa(product_prod(int,int),int,product_fst(int,int),Y2),aa(product_prod(int,int),int,product_snd(int,int),Y2)) )
            & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Y2))
            & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Y2),aa(product_prod(int,int),int,product_snd(int,int),Y2)) )
         => ( Y2 = X3 ) ) ) ).

% quotient_of_unique
tff(fact_7319_gcd__nat_Opelims,axiom,
    ! [Xa: nat,Xaa: nat,Ya: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),Xaa) = Ya )
     => ( aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Xaa))
       => ~ ( ( Ya = $ite(Xaa = zero_zero(nat),Xa,aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xaa),modulo_modulo(nat,Xa,Xaa))) )
           => ~ aa(product_prod(nat,nat),$o,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_7320_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),A3: list(A),B3: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,R),A3),B3)
    <=> ? [Z5: list(product_prod(A,B))] :
          ( member(list(product_prod(A,B)),Z5,aa(fun(list(product_prod(A,B)),$o),set(list(product_prod(A,B))),collect(list(product_prod(A,B))),aTP_Lamp_ahp(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),R)))
          & ( map(product_prod(A,B),A,product_fst(A,B),Z5) = A3 )
          & ( map(product_prod(A,B),B,product_snd(A,B),Z5) = B3 ) ) ) ).

% list.in_rel
tff(fact_7321_coprime__Suc__0__left,axiom,
    ! [Na: nat] : algebr8660921524188924756oprime(nat,aa(nat,nat,suc,zero_zero(nat)),Na) ).

% coprime_Suc_0_left
tff(fact_7322_coprime__Suc__0__right,axiom,
    ! [Na: nat] : algebr8660921524188924756oprime(nat,Na,aa(nat,nat,suc,zero_zero(nat))) ).

% coprime_Suc_0_right
tff(fact_7323_list_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,$o)),Ra: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R),Ra)
     => aa(fun(list(A),fun(list(B),$o)),$o,aa(fun(list(A),fun(list(B),$o)),fun(fun(list(A),fun(list(B),$o)),$o),ord_less_eq(fun(list(A),fun(list(B),$o))),list_all2(A,B,R)),list_all2(A,B,Ra)) ) ).

% list.rel_mono
tff(fact_7324_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys2: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys2)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys2) )
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,B,nth(B,Ys2),I4)) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_7325_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A3: list(A),B3: list(B),P: fun(A,fun(B,$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),A3) = aa(list(B),nat,size_size(list(B)),B3) )
     => ( ! [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),A3))
           => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,A3),N2)),aa(nat,B,nth(B,B3),N2)) )
       => aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),A3),B3) ) ) ).

% list_all2_all_nthI
tff(fact_7326_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys2: list(B),P3: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P3),aa(list(B),nat,size_size(list(B)),Ys2))
       => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3)) ) ) ).

% list_all2_nthD2
tff(fact_7327_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xs: list(A),Ys2: list(B),P3: nat] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),list_all2(A,B,P),Xs),Ys2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),P3),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(B,$o,aa(A,fun(B,$o),P,aa(nat,A,nth(A,Xs),P3)),aa(nat,B,nth(B,Ys2),P3)) ) ) ).

% list_all2_nthD
tff(fact_7328_coprime__diff__one__right__nat,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => algebr8660921524188924756oprime(nat,Na,aa(nat,nat,minus_minus(nat,Na),one_one(nat))) ) ).

% coprime_diff_one_right_nat
tff(fact_7329_coprime__diff__one__left__nat,axiom,
    ! [Na: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Na)
     => algebr8660921524188924756oprime(nat,aa(nat,nat,minus_minus(nat,Na),one_one(nat)),Na) ) ).

% coprime_diff_one_left_nat
tff(fact_7330_sum__list__transfer,axiom,
    ! [A: $tType,B: $tType] :
      ( ( monoid_add(B)
        & monoid_add(A) )
     => ! [A4: fun(A,fun(B,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A4,zero_zero(A)),zero_zero(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),plus_plus(A)),plus_plus(B))
           => aa(fun(list(B),B),$o,aa(fun(list(A),A),fun(fun(list(B),B),$o),bNF_rel_fun(list(A),list(B),A,B,list_all2(A,B,A4),A4),groups8242544230860333062m_list(A)),groups8242544230860333062m_list(B)) ) ) ) ).

% sum_list_transfer
tff(fact_7331_Rats__abs__nat__div__natE,axiom,
    ! [Xa: real] :
      ( member(real,Xa,field_char_0_Rats(real))
     => ~ ! [M4: nat,N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => ( ( aa(real,real,abs_abs(real),Xa) = 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_7332_horner__sum__transfer,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( comm_semiring_0(B)
        & comm_semiring_0(A) )
     => ! [A4: fun(A,fun(B,$o)),B2: fun(C,fun(D,$o))] :
          ( aa(B,$o,aa(A,fun(B,$o),A4,zero_zero(A)),zero_zero(B))
         => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),plus_plus(A)),plus_plus(B))
           => ( aa(fun(B,fun(B,B)),$o,aa(fun(A,fun(A,A)),fun(fun(B,fun(B,B)),$o),bNF_rel_fun(A,B,fun(A,A),fun(B,B),A4,bNF_rel_fun(A,B,A,B,A4,A4)),times_times(A)),times_times(B))
             => aa(fun(fun(D,B),fun(B,fun(list(D),B))),$o,aa(fun(fun(C,A),fun(A,fun(list(C),A))),fun(fun(fun(D,B),fun(B,fun(list(D),B))),$o),bNF_rel_fun(fun(C,A),fun(D,B),fun(A,fun(list(C),A)),fun(B,fun(list(D),B)),bNF_rel_fun(C,D,A,B,B2,A4),bNF_rel_fun(A,B,fun(list(C),A),fun(list(D),B),A4,bNF_rel_fun(list(C),list(D),A,B,list_all2(C,D,B2),A4))),groups4207007520872428315er_sum(C,A)),groups4207007520872428315er_sum(D,B)) ) ) ) ) ).

% horner_sum_transfer
tff(fact_7333_relImage__Gr,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),A4: set(A),F2: fun(A,B)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))
     => ( bNF_Gr4221423524335903396lImage(A,B,R,F2) = relcomp(B,A,B,converse(A,B,bNF_Gr(A,B,A4,F2)),relcomp(A,A,B,R,bNF_Gr(A,B,A4,F2))) ) ) ).

% relImage_Gr
tff(fact_7334_relInvImage__Gr,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(A,A)),B2: set(A),A4: set(B),F2: fun(B,A)] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2)))
     => ( bNF_Gr7122648621184425601vImage(B,A,A4,R,F2) = relcomp(B,A,B,bNF_Gr(B,A,A4,F2),relcomp(A,A,B,R,converse(B,A,bNF_Gr(B,A,A4,F2)))) ) ) ).

% relInvImage_Gr
tff(fact_7335_converse__inject,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S3: set(product_prod(B,A))] :
      ( ( converse(B,A,R2) = converse(B,A,S3) )
    <=> ( R2 = S3 ) ) ).

% converse_inject
tff(fact_7336_converse__converse,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : converse(B,A,converse(A,B,R2)) = R2 ).

% converse_converse
tff(fact_7337_converse__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),converse(B,A,R2))
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,B3),A3),R2) ) ).

% converse_iff
tff(fact_7338_Field__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),converse(A,A,R2)) = aa(set(product_prod(A,A)),set(A),field2(A),R2) ).

% Field_converse
tff(fact_7339_converse__mono,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S3: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),converse(B,A,R2)),converse(B,A,S3))
    <=> aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),R2),S3) ) ).

% converse_mono
tff(fact_7340_converse__empty,axiom,
    ! [B: $tType,A: $tType] : converse(B,A,bot_bot(set(product_prod(B,A)))) = bot_bot(set(product_prod(A,B))) ).

% converse_empty
tff(fact_7341_converse__Id,axiom,
    ! [A: $tType] : converse(A,A,id2(A)) = id2(A) ).

% converse_Id
tff(fact_7342_refl__on__converse,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A4,converse(A,A,R2))
    <=> refl_on(A,A4,R2) ) ).

% refl_on_converse
tff(fact_7343_finite__converse,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),converse(B,A,R2))
    <=> aa(set(product_prod(B,A)),$o,finite_finite2(product_prod(B,A)),R2) ) ).

% finite_converse
tff(fact_7344_total__on__converse,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( total_on(A,A4,converse(A,A,R2))
    <=> total_on(A,A4,R2) ) ).

% total_on_converse
tff(fact_7345_antisym__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisym(A,converse(A,A,R2))
    <=> antisym(A,R2) ) ).

% antisym_converse
tff(fact_7346_converse__UNIV,axiom,
    ! [B: $tType,A: $tType] : converse(B,A,top_top(set(product_prod(B,A)))) = top_top(set(product_prod(A,B))) ).

% converse_UNIV
tff(fact_7347_converse__Id__on,axiom,
    ! [A: $tType,A4: set(A)] : converse(A,A,id_on(A,A4)) = id_on(A,A4) ).

% converse_Id_on
tff(fact_7348_card__inverse,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A))] : aa(set(product_prod(A,B)),nat,finite_card(product_prod(A,B)),converse(B,A,R)) = aa(set(product_prod(B,A)),nat,finite_card(product_prod(B,A)),R) ).

% card_inverse
tff(fact_7349_converse__relcomp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(B,C)),S3: set(product_prod(C,A))] : converse(B,A,relcomp(B,C,A,R2,S3)) = relcomp(A,C,B,converse(C,A,S3),converse(B,C,R2)) ).

% converse_relcomp
tff(fact_7350_converse__Un,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S3: set(product_prod(B,A))] : converse(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),R2),S3)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),converse(B,A,R2)),converse(B,A,S3)) ).

% converse_Un
tff(fact_7351_converseI,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),R2)
     => member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,B3),A3),converse(A,B,R2)) ) ).

% converseI
tff(fact_7352_converseE,axiom,
    ! [A: $tType,B: $tType,Yx: product_prod(A,B),R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),Yx,converse(B,A,R2))
     => ~ ! [X3: B,Y: A] :
            ( ( Yx = aa(B,product_prod(A,B),product_Pair(A,B,Y),X3) )
           => ~ member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,X3),Y),R2) ) ) ).

% converseE
tff(fact_7353_converseD,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),converse(B,A,R2))
     => member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,B3),A3),R2) ) ).

% converseD
tff(fact_7354_converse_Osimps,axiom,
    ! [A: $tType,B: $tType,A12: A,A23: B,R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A12),A23),converse(B,A,R2))
    <=> ? [A9: B,B7: A] :
          ( ( A12 = B7 )
          & ( A23 = A9 )
          & member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,A9),B7),R2) ) ) ).

% converse.simps
tff(fact_7355_converse_Ocases,axiom,
    ! [A: $tType,B: $tType,A12: A,A23: B,R2: set(product_prod(B,A))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A12),A23),converse(B,A,R2))
     => member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,A23),A12),R2) ) ).

% converse.cases
tff(fact_7356_converse__UNION,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R2),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_ahq(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R2)),S)) ).

% converse_UNION
tff(fact_7357_converse__unfold,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : converse(B,A,R2) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_ahr(set(product_prod(B,A)),fun(A,fun(B,$o)),R2))) ).

% converse_unfold
tff(fact_7358_converse__INTER,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: fun(C,set(product_prod(B,A))),S: set(C)] : converse(B,A,aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Inf_Inf(set(product_prod(B,A))),aa(set(C),set(set(product_prod(B,A))),image2(C,set(product_prod(B,A)),R2),S))) = aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Inf_Inf(set(product_prod(A,B))),aa(set(C),set(set(product_prod(A,B))),image2(C,set(product_prod(A,B)),aTP_Lamp_ahq(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),R2)),S)) ).

% converse_INTER
tff(fact_7359_converse__Int,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),S3: set(product_prod(B,A))] : converse(B,A,aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),R2),S3)) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),converse(B,A,R2)),converse(B,A,S3)) ).

% converse_Int
tff(fact_7360_converse__subset__swap,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(B,A))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),converse(B,A,S3))
    <=> aa(set(product_prod(B,A)),$o,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),$o),ord_less_eq(set(product_prod(B,A))),converse(A,B,R2)),S3) ) ).

% converse_subset_swap
tff(fact_7361_Image__subset__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,A)),A4: set(B),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,R2),A4)),B2)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),A4),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,converse(B,A,R2)),aa(set(A),set(A),uminus_uminus(set(A)),B2)))) ) ).

% Image_subset_eq
tff(fact_7362_refl__on__comp__subset,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] :
      ( refl_on(A,A4,R2)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),relcomp(A,A,A,converse(A,A,R2),R2)) ) ).

% refl_on_comp_subset
tff(fact_7363_Image__INT__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(B,A)),A4: set(C),B2: fun(C,set(B))] :
      ( single_valued(A,B,converse(B,A,R2))
     => ( ( A4 != bot_bot(set(C)) )
       => ( aa(set(B),set(A),image(B,A,R2),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image2(C,set(B),B2),A4))) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ach(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),R2),B2)),A4)) ) ) ) ).

% Image_INT_eq
tff(fact_7364_trans__wf__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => ( wf(A,R2)
      <=> ! [A9: A] : wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,aa(set(A),set(A),image(A,A,converse(A,A,R2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A9),bot_bot(set(A)))),aa(A,fun(A,set(A)),aTP_Lamp_ahs(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A9)))) ) ) ).

% trans_wf_iff
tff(fact_7365_trans__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,converse(A,A,R2))
    <=> trans(A,R2) ) ).

% trans_converse
tff(fact_7366_trans__O__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R2,R2)),R2) ) ).

% trans_O_subset
tff(fact_7367_single__valued__subset,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S3)
     => ( single_valued(A,B,S3)
       => single_valued(A,B,R2) ) ) ).

% single_valued_subset
tff(fact_7368_trans__Id,axiom,
    ! [A: $tType] : trans(A,id2(A)) ).

% trans_Id
tff(fact_7369_single__valued__Id,axiom,
    ! [A: $tType] : single_valued(A,A,id2(A)) ).

% single_valued_Id
tff(fact_7370_single__valued__Id__on,axiom,
    ! [A: $tType,A4: set(A)] : single_valued(A,A,id_on(A,A4)) ).

% single_valued_Id_on
tff(fact_7371_trans__Id__on,axiom,
    ! [A: $tType,A4: set(A)] : trans(A,id_on(A,A4)) ).

% trans_Id_on
tff(fact_7372_trans__Int,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( trans(A,R2)
     => ( trans(A,S3)
       => trans(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),S3)) ) ) ).

% trans_Int
tff(fact_7373_trans__INTER,axiom,
    ! [B: $tType,A: $tType,S: set(A),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [X3: A] :
          ( member(A,X3,S)
         => trans(B,aa(A,set(product_prod(B,B)),R2,X3)) )
     => trans(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).

% trans_INTER
tff(fact_7374_transD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Ya: A,Z: A] :
      ( trans(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Ya),Z),R2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Z),R2) ) ) ) ).

% transD
tff(fact_7375_transE,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Ya: A,Z: A] :
      ( trans(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Ya),Z),R2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Z),R2) ) ) ) ).

% transE
tff(fact_7376_transI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [X3: A,Y: A,Z2: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y),Z2),R2)
           => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X3),Z2),R2) ) )
     => trans(A,R2) ) ).

% transI
tff(fact_7377_trans__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
    <=> ! [X: A,Y4: A,Z5: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4),R2)
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z5),R2)
           => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Z5),R2) ) ) ) ).

% trans_def
tff(fact_7378_single__valuedD,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B)),Xa: A,Ya: B,Z: B] :
      ( single_valued(A,B,R2)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xa),Ya),R2)
       => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Xa),Z),R2)
         => ( Ya = Z ) ) ) ) ).

% single_valuedD
tff(fact_7379_single__valuedI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( ! [X3: A,Y: B,Z2: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y),R2)
         => ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X3),Z2),R2)
           => ( Y = Z2 ) ) )
     => single_valued(A,B,R2) ) ).

% single_valuedI
tff(fact_7380_single__valued__def,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,B))] :
      ( single_valued(A,B,R2)
    <=> ! [X: A,Y4: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X),Y4),R2)
         => ! [Z5: B] :
              ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,X),Z5),R2)
             => ( Y4 = Z5 ) ) ) ) ).

% single_valued_def
tff(fact_7381_trans__empty,axiom,
    ! [A: $tType] : trans(A,bot_bot(set(product_prod(A,A)))) ).

% trans_empty
tff(fact_7382_single__valued__empty,axiom,
    ! [B: $tType,A: $tType] : single_valued(A,B,bot_bot(set(product_prod(A,B)))) ).

% single_valued_empty
tff(fact_7383_single__valued__relcomp,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(B,C))] :
      ( single_valued(A,B,R2)
     => ( single_valued(B,C,S3)
       => single_valued(A,C,relcomp(A,B,C,R2,S3)) ) ) ).

% single_valued_relcomp
tff(fact_7384_under__incr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( trans(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_under(A,R2,A3)),order_under(A,R2,B3)) ) ) ).

% under_incr
tff(fact_7385_trans__singleton,axiom,
    ! [A: $tType,A3: A] : trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A3),A3)),bot_bot(set(product_prod(A,A))))) ).

% trans_singleton
tff(fact_7386_trans__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
     => ( antisym(A,R2)
       => trans(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),R2),id2(A))) ) ) ).

% trans_diff_Id
tff(fact_7387_trans__join,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( trans(A,R2)
    <=> ! [X: product_prod(A,A)] :
          ( member(product_prod(A,A),X,R2)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_ahu(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)),X) ) ) ).

% trans_join
tff(fact_7388_underS__incr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A,B3: A] :
      ( trans(A,R2)
     => ( antisym(A,R2)
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_underS(A,R2,A3)),order_underS(A,R2,B3)) ) ) ) ).

% underS_incr
tff(fact_7389_Image__Int__eq,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),A4: set(B),B2: set(B)] :
      ( single_valued(A,B,converse(B,A,R))
     => ( aa(set(B),set(A),image(B,A,R),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A4),B2)) = 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,R),A4)),aa(set(B),set(A),image(B,A,R),B2)) ) ) ).

% Image_Int_eq
tff(fact_7390_wf__finite__segments,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
     => ( trans(A,R2)
       => ( ! [X3: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_ahv(set(product_prod(A,A)),fun(A,fun(A,$o)),R2),X3)))
         => wf(A,R2) ) ) ) ).

% wf_finite_segments
tff(fact_7391_Rat_Opositive_Orsp,axiom,
    aa(fun(product_prod(int,int),$o),$o,aa(fun(product_prod(int,int),$o),fun(fun(product_prod(int,int),$o),$o),bNF_rel_fun(product_prod(int,int),product_prod(int,int),$o,$o,ratrel,fequal($o)),aTP_Lamp_ahl(product_prod(int,int),$o)),aTP_Lamp_ahl(product_prod(int,int),$o)) ).

% Rat.positive.rsp
tff(fact_7392_irrefl__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [A9: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A9),A9),R2) ) ).

% irrefl_def
tff(fact_7393_irreflI,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( ! [A5: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A5),A5),R)
     => irrefl(A,R) ) ).

% irreflI
tff(fact_7394_irrefl__diff__Id,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : irrefl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),R2),id2(A))) ).

% irrefl_diff_Id
tff(fact_7395_irrefl__distinct,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( irrefl(A,R2)
    <=> ! [X: product_prod(A,A)] :
          ( member(product_prod(A,A),X,R2)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_ahw(A,fun(A,$o))),X) ) ) ).

% irrefl_distinct
tff(fact_7396_Rat_Opositive_Oabs__eq,axiom,
    ! [Xa: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aa(product_prod(int,int),fun(product_prod(int,int),$o),ratrel,Xa),Xa)
     => ( aa(rat,$o,positive,abs_Rat(Xa))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Xa)),aa(product_prod(int,int),int,product_snd(int,int),Xa))) ) ) ).

% Rat.positive.abs_eq
tff(fact_7397_irreflp__irrefl__eq,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irreflp(A,aTP_Lamp_ahx(set(product_prod(A,A)),fun(A,fun(A,$o)),R))
    <=> irrefl(A,R) ) ).

% irreflp_irrefl_eq
tff(fact_7398_irreflp__def,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( irreflp(A,R)
    <=> ! [A9: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,A9),A9) ) ).

% irreflp_def
tff(fact_7399_irreflpI,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( ! [A5: A] : ~ aa(A,$o,aa(A,fun(A,$o),R,A5),A5)
     => irreflp(A,R) ) ).

% irreflpI
tff(fact_7400_irreflp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,ord_less(A)) ) ).

% irreflp_less
tff(fact_7401_irreflp__greater,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => irreflp(A,aTP_Lamp_ahy(A,fun(A,$o))) ) ).

% irreflp_greater
tff(fact_7402_le__prod__filterI,axiom,
    ! [A: $tType,B: $tType,F3: filter(product_prod(A,B)),A4: filter(A),B2: filter(B)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),filtermap(product_prod(A,B),A),product_fst(A,B)),F3)),A4)
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),filtermap(product_prod(A,B),B),product_snd(A,B)),F3)),B2)
       => aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),F3),prod_filter(A,B,A4,B2)) ) ) ).

% le_prod_filterI
tff(fact_7403_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F2: fun(nat,real),M5: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahz(fun(nat,real),fun(nat,fun(nat,real)),F2),M5),at_top(nat))
     => ( ! [M4: nat,N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),M4)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,N2)),aa(nat,real,F2,M4)) ) )
       => topolo6863149650580417670ergent(real,F2) ) ) ).

% Bseq_monoseq_convergent'_dec
tff(fact_7404_filtermap__id_H,axiom,
    ! [A: $tType,X2: filter(A)] : aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_jq(A,A)),X2) = X2 ).

% filtermap_id'
tff(fact_7405_filtermap__bot,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),bot_bot(filter(B))) = bot_bot(filter(A)) ).

% filtermap_bot
tff(fact_7406_filtermap__principal,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),A4: set(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),principal(B,A4)) = principal(A,aa(set(B),set(A),image2(B,A,F2),A4)) ).

% filtermap_principal
tff(fact_7407_filtermap__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),F3: fun(C,filter(B)),B2: set(C)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(set(filter(B)),filter(B),complete_Sup_Sup(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F3),B2))) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aia(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F3)),B2)) ).

% filtermap_SUP
tff(fact_7408_filtermap__bot__iff,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),F3: filter(B)] :
      ( ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F3) = bot_bot(filter(A)) )
    <=> ( F3 = bot_bot(filter(B)) ) ) ).

% filtermap_bot_iff
tff(fact_7409_filtermap__sup,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),F1: filter(B),F22: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),sup_sup(filter(B)),F1),F22)) = aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),sup_sup(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F1)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F22)) ).

% filtermap_sup
tff(fact_7410_prod__filter__assoc,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: filter(A),G2: filter(B),H6: filter(C)] : prod_filter(product_prod(A,B),C,prod_filter(A,B,F3,G2),H6) = aa(filter(product_prod(A,product_prod(B,C))),filter(product_prod(product_prod(A,B),C)),aa(fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),fun(filter(product_prod(A,product_prod(B,C))),filter(product_prod(product_prod(A,B),C))),filtermap(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),aa(fun(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),fun(product_prod(A,product_prod(B,C)),product_prod(product_prod(A,B),C)),product_case_prod(A,product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_aic(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))))),prod_filter(A,product_prod(B,C),F3,prod_filter(B,C,G2,H6))) ).

% prod_filter_assoc
tff(fact_7411_convergent__mult__const__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qr(A,fun(fun(nat,A),fun(nat,A)),C3),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_iff
tff(fact_7412_convergent__mult__const__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C3: A,F2: fun(nat,A)] :
          ( ( C3 != zero_zero(A) )
         => ( topolo6863149650580417670ergent(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_qq(A,fun(fun(nat,A),fun(nat,A)),C3),F2))
          <=> topolo6863149650580417670ergent(A,F2) ) ) ) ).

% convergent_mult_const_right_iff
tff(fact_7413_eventually__filtermap,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A),F3: filter(B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F3))
    <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_sr(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2)),F3) ) ).

% eventually_filtermap
tff(fact_7414_filterlim__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(A,B),F1: filter(B),G: fun(C,A),F22: filter(C)] :
      ( filterlim(A,B,F2,F1,aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),G),F22))
    <=> filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pg(fun(A,B),fun(fun(C,A),fun(C,B)),F2),G),F1,F22) ) ).

% filterlim_filtermap
tff(fact_7415_filtermap__ident,axiom,
    ! [A: $tType,F3: filter(A)] : aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_jq(A,A)),F3) = F3 ).

% filtermap_ident
tff(fact_7416_filtermap__filtermap,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),G: fun(C,B),F3: filter(C)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(filter(C),filter(B),aa(fun(C,B),fun(filter(C),filter(B)),filtermap(C,B),G),F3)) = aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),aa(fun(C,B),fun(C,A),aTP_Lamp_ka(fun(B,A),fun(fun(C,B),fun(C,A)),F2),G)),F3) ).

% filtermap_filtermap
tff(fact_7417_prod__filtermap2,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: filter(A),G: fun(C,B),G2: filter(C)] : prod_filter(A,B,F3,aa(filter(C),filter(B),aa(fun(C,B),fun(filter(C),filter(B)),filtermap(C,B),G),G2)) = aa(filter(product_prod(A,C)),filter(product_prod(A,B)),aa(fun(product_prod(A,C),product_prod(A,B)),fun(filter(product_prod(A,C)),filter(product_prod(A,B))),filtermap(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A,G)),prod_filter(A,C,F3,G2)) ).

% prod_filtermap2
tff(fact_7418_map__filter__on__UNIV,axiom,
    ! [B: $tType,A: $tType] : map_filter_on(A,B,top_top(set(A))) = filtermap(A,B) ).

% map_filter_on_UNIV
tff(fact_7419_filtermap__fun__inverse,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),F3: filter(B),G2: filter(A),F2: fun(B,A)] :
      ( filterlim(A,B,G,F3,G2)
     => ( filterlim(B,A,F2,G2,F3)
       => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(B,A),fun(A,$o),aTP_Lamp_aid(fun(A,B),fun(fun(B,A),fun(A,$o)),G),F2)),G2)
         => ( aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F3) = G2 ) ) ) ) ).

% filtermap_fun_inverse
tff(fact_7420_filtermap__eq__strong,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(A),G2: filter(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( ( aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F3) = aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),G2) )
      <=> ( F3 = G2 ) ) ) ).

% filtermap_eq_strong
tff(fact_7421_filtermap__nhds__times,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C3: A,A3: A] :
          ( ( C3 != zero_zero(A) )
         => ( aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aa(A,fun(A,A),times_times(A),C3)),topolo7230453075368039082e_nhds(A,A3)) = topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),A3)) ) ) ) ).

% filtermap_nhds_times
tff(fact_7422_filtercomap__filtermap,axiom,
    ! [B: $tType,A: $tType,F3: filter(A),F2: fun(A,B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),filtercomap(A,B,F2,aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F3))) ).

% filtercomap_filtermap
tff(fact_7423_filtermap__filtercomap,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),F3: filter(A)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),filtercomap(B,A,F2,F3))),F3) ).

% filtermap_filtercomap
tff(fact_7424_filtermap__le__iff__le__filtercomap,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),F3: filter(B),G2: filter(A)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F3)),G2)
    <=> aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),F3),filtercomap(B,A,F2,G2)) ) ).

% filtermap_le_iff_le_filtercomap
tff(fact_7425_filterlim__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,B),F22: filter(B),F1: filter(A)] :
      ( filterlim(A,B,F2,F22,F1)
    <=> aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F1)),F22) ) ).

% filterlim_def
tff(fact_7426_lim__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F2: fun(nat,A),Xa: A] :
          ( topolo6863149650580417670ergent(A,F2)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F2,N2)),Xa)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),topolo3827282254853284352ce_Lim(nat,A,at_top(nat),F2)),Xa) ) ) ) ).

% lim_le
tff(fact_7427_filtermap__mono,axiom,
    ! [B: $tType,A: $tType,F3: filter(A),F8: filter(A),F2: fun(A,B)] :
      ( aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),F8)
     => aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F3)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F8)) ) ).

% filtermap_mono
tff(fact_7428_filtermap__inf,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),F1: filter(B),F22: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),F1),F22))),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F1)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F22))) ).

% filtermap_inf
tff(fact_7429_filtermap__Pair,axiom,
    ! [A: $tType,B: $tType,C: $tType,F2: fun(C,A),G: fun(C,B),F3: filter(C)] : aa(filter(product_prod(A,B)),$o,aa(filter(product_prod(A,B)),fun(filter(product_prod(A,B)),$o),ord_less_eq(filter(product_prod(A,B))),aa(filter(C),filter(product_prod(A,B)),aa(fun(C,product_prod(A,B)),fun(filter(C),filter(product_prod(A,B))),filtermap(C,product_prod(A,B)),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_afx(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),F2),G)),F3)),prod_filter(A,B,aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),F2),F3),aa(filter(C),filter(B),aa(fun(C,B),fun(filter(C),filter(B)),filtermap(C,B),G),F3))) ).

% filtermap_Pair
tff(fact_7430_filtermap__def,axiom,
    ! [A: $tType,B: $tType,F2: fun(B,A),F3: filter(B)] : aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F3) = abs_filter(A,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_aif(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),F2),F3)) ).

% filtermap_def
tff(fact_7431_filtermap__mono__strong,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),F3: filter(A),G2: filter(A)] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( aa(filter(B),$o,aa(filter(B),fun(filter(B),$o),ord_less_eq(filter(B)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),F3)),aa(filter(A),filter(B),aa(fun(A,B),fun(filter(A),filter(B)),filtermap(A,B),F2),G2))
      <=> aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),F3),G2) ) ) ).

% filtermap_mono_strong
tff(fact_7432_filtermap__fst__prod__filter,axiom,
    ! [B: $tType,A: $tType,A4: filter(A),B2: filter(B)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),filtermap(product_prod(A,B),A),product_fst(A,B)),prod_filter(A,B,A4,B2))),A4) ).

% filtermap_fst_prod_filter
tff(fact_7433_filtermap__snd__prod__filter,axiom,
    ! [B: $tType,A: $tType,A4: filter(B),B2: filter(A)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(product_prod(B,A)),filter(A),aa(fun(product_prod(B,A),A),fun(filter(product_prod(B,A)),filter(A)),filtermap(product_prod(B,A),A),product_snd(B,A)),prod_filter(B,A,A4,B2))),B2) ).

% filtermap_snd_prod_filter
tff(fact_7434_filtermap__INF,axiom,
    ! [B: $tType,A: $tType,C: $tType,F2: fun(B,A),F3: fun(C,filter(B)),B2: set(C)] : aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(C),set(filter(B)),image2(C,filter(B),F3),B2)))),aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(C),set(filter(A)),image2(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_aia(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),F2),F3)),B2))) ).

% filtermap_INF
tff(fact_7435_at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A] : topolo174197925503356063within(A,A3,top_top(set(A))) = aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aTP_Lamp_aig(A,fun(A,A),A3)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% at_to_0
tff(fact_7436_Bseq__mono__convergent,axiom,
    ! [X4: fun(nat,real)] :
      ( bfun(nat,real,X4,at_top(nat))
     => ( ! [M4: nat,N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,X4,M4)),aa(nat,real,X4,N2)) )
       => topolo6863149650580417670ergent(real,X4) ) ) ).

% Bseq_mono_convergent
tff(fact_7437_convergent__realpow,axiom,
    ! [Xa: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xa)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),one_one(real))
       => topolo6863149650580417670ergent(real,power_power(real,Xa)) ) ) ).

% convergent_realpow
tff(fact_7438_filterlim__INF__INF,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,J4: set(A),I5: set(B),F2: fun(D,C),F3: fun(B,filter(D)),G2: fun(A,filter(C))] :
      ( ! [M4: A] :
          ( member(A,M4,J4)
         => ? [X2: B] :
              ( member(B,X2,I5)
              & aa(filter(C),$o,aa(filter(C),fun(filter(C),$o),ord_less_eq(filter(C)),aa(filter(D),filter(C),aa(fun(D,C),fun(filter(D),filter(C)),filtermap(D,C),F2),aa(B,filter(D),F3,X2))),aa(A,filter(C),G2,M4)) ) )
     => filterlim(D,C,F2,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(A),set(filter(C)),image2(A,filter(C),G2),J4)),aa(set(filter(D)),filter(D),complete_Inf_Inf(filter(D)),aa(set(B),set(filter(D)),image2(B,filter(D),F3),I5))) ) ).

% filterlim_INF_INF
tff(fact_7439_filtermap__times__pos__at__right,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [C3: A,P3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),C3)
         => ( aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),aa(A,fun(A,A),times_times(A),C3)),topolo174197925503356063within(A,P3,aa(A,set(A),set_ord_greaterThan(A),P3))) = topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),times_times(A),C3),P3),aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),times_times(A),C3),P3))) ) ) ) ).

% filtermap_times_pos_at_right
tff(fact_7440_at__to__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ( topolo174197925503356063within(A,zero_zero(A),top_top(set(A))) = aa(filter(A),filter(A),aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),inverse_inverse(A)),at_infinity(A)) ) ) ).

% at_to_infinity
tff(fact_7441_cauchy__filter__metric__filtermap,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F2: fun(B,A),F3: filter(B)] :
          ( topolo6773858410816713723filter(A,aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),F2),F3))
        <=> ! [E3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),E3)
             => ? [P5: fun(B,$o)] :
                  ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),P5),F3)
                  & ! [X: B,Y4: B] :
                      ( ( aa(B,$o,P5,X)
                        & aa(B,$o,P5,Y4) )
                     => aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,aa(B,A,F2,X),aa(B,A,F2,Y4))),E3) ) ) ) ) ) ).

% cauchy_filter_metric_filtermap
tff(fact_7442_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F2: fun(nat,real),M5: nat] :
      ( bfun(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahz(fun(nat,real),fun(nat,fun(nat,real)),F2),M5),at_top(nat))
     => ( ! [M4: nat,N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M5),M4)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M4),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F2,M4)),aa(nat,real,F2,N2)) ) )
       => topolo6863149650580417670ergent(real,F2) ) ) ).

% Bseq_monoseq_convergent'_inc
tff(fact_7443_sorted__insort__insert__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F2: fun(B,A),Xs: list(B),Xa: 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,Xa,Xs))) ) ) ).

% sorted_insort_insert_key
tff(fact_7444_pair__lessI2,axiom,
    ! [A3: nat,B3: nat,S3: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),S3),Ta)
       => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S3)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B3),Ta)),fun_pair_less) ) ) ).

% pair_lessI2
tff(fact_7445_pair__less__iff1,axiom,
    ! [Xa: nat,Ya: nat,Z: nat] :
      ( member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Ya)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Z)),fun_pair_less)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Ya),Z) ) ).

% pair_less_iff1
tff(fact_7446_filtermap__sequentually__ne__bot,axiom,
    ! [A: $tType,F2: fun(nat,A)] : aa(filter(nat),filter(A),aa(fun(nat,A),fun(filter(nat),filter(A)),filtermap(nat,A),F2),at_top(nat)) != bot_bot(filter(A)) ).

% filtermap_sequentually_ne_bot
tff(fact_7447_filtermap__image__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(filter(set(A)),filter(set(B)),aa(fun(set(A),set(B)),fun(filter(set(A)),filter(set(B))),filtermap(set(A),set(B)),image2(A,B,F2)),finite5375528669736107172at_top(A,A4)) = finite5375528669736107172at_top(B,aa(set(A),set(B),image2(A,B,F2),A4)) ) ) ).

% filtermap_image_finite_subsets_at_top
tff(fact_7448_pair__lessI1,axiom,
    ! [A3: nat,B3: nat,S3: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S3)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B3),Ta)),fun_pair_less) ) ).

% pair_lessI1
tff(fact_7449_set__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xa: A,Xs: list(A)] : aa(list(A),set(A),set2(A),linord329482645794927042rt_key(A,A,aTP_Lamp_jn(A,A),Xa,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(list(A),set(A),set2(A),Xs)) ) ).

% set_insort_insert
tff(fact_7450_sorted__insort__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Xa: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),linord329482645794927042rt_key(A,A,aTP_Lamp_jn(A,A),Xa,Xs)) ) ) ).

% sorted_insort_insert
tff(fact_7451_prod__filter__principal__singleton,axiom,
    ! [A: $tType,B: $tType,Xa: A,F3: filter(B)] : prod_filter(A,B,principal(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))),F3) = aa(filter(B),filter(product_prod(A,B)),aa(fun(B,product_prod(A,B)),fun(filter(B),filter(product_prod(A,B))),filtermap(B,product_prod(A,B)),product_Pair(A,B,Xa)),F3) ).

% prod_filter_principal_singleton
tff(fact_7452_prod__filter__principal__singleton2,axiom,
    ! [B: $tType,A: $tType,F3: filter(A),Xa: B] : prod_filter(A,B,F3,principal(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Xa),bot_bot(set(B))))) = aa(filter(A),filter(product_prod(A,B)),aa(fun(A,product_prod(A,B)),fun(filter(A),filter(product_prod(A,B))),filtermap(A,product_prod(A,B)),aTP_Lamp_aih(B,fun(A,product_prod(A,B)),Xa)),F3) ).

% prod_filter_principal_singleton2
tff(fact_7453_pair__leqI2,axiom,
    ! [A3: nat,B3: nat,S3: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A3),B3)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),S3),Ta)
       => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S3)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B3),Ta)),fun_pair_leq) ) ) ).

% pair_leqI2
tff(fact_7454_pair__leqI1,axiom,
    ! [A3: nat,B3: nat,S3: nat,Ta: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A3),B3)
     => member(product_prod(product_prod(nat,nat),product_prod(nat,nat)),aa(product_prod(nat,nat),product_prod(product_prod(nat,nat),product_prod(nat,nat)),product_Pair(product_prod(nat,nat),product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,A3),S3)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,B3),Ta)),fun_pair_leq) ) ).

% pair_leqI1
tff(fact_7455_admissible__chfin,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,$o)] :
          ( ! [S2: set(A)] :
              ( comple1602240252501008431_chain(A,ord_less_eq(A),S2)
             => aa(set(A),$o,finite_finite2(A),S2) )
         => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P) ) ) ).

% admissible_chfin
tff(fact_7456_bot_Oordering__top__axioms,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ordering_top(A,aTP_Lamp_aii(A,fun(A,$o)),aTP_Lamp_aij(A,fun(A,$o)),bot_bot(A)) ) ).

% bot.ordering_top_axioms
tff(fact_7457_ccpo_OadmissibleD,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),P: fun(A,$o),A4: set(A)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P)
     => ( comple1602240252501008431_chain(A,Ord,A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( member(A,X3,A4)
               => aa(A,$o,P,X3) )
           => aa(A,$o,P,aa(set(A),A,Lub,A4)) ) ) ) ) ).

% ccpo.admissibleD
tff(fact_7458_ccpo_OadmissibleI,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,$o)),P: fun(A,$o),Lub: fun(set(A),A)] :
      ( ! [A7: set(A)] :
          ( comple1602240252501008431_chain(A,Ord,A7)
         => ( ( A7 != bot_bot(set(A)) )
           => ( ! [X2: A] :
                  ( member(A,X2,A7)
                 => aa(A,$o,P,X2) )
             => aa(A,$o,P,aa(set(A),A,Lub,A7)) ) ) )
     => comple1908693960933563346ssible(A,Lub,Ord,P) ) ).

% ccpo.admissibleI
tff(fact_7459_ccpo_Oadmissible__def,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),P: fun(A,$o)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P)
    <=> ! [A8: set(A)] :
          ( comple1602240252501008431_chain(A,Ord,A8)
         => ( ( A8 != bot_bot(set(A)) )
           => ( ! [X: A] :
                  ( member(A,X,A8)
                 => aa(A,$o,P,X) )
             => aa(A,$o,P,aa(set(A),A,Lub,A8)) ) ) ) ) ).

% ccpo.admissible_def
tff(fact_7460_ordering__top_Oextremum__uniqueI,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Top),A3)
       => ( A3 = Top ) ) ) ).

% ordering_top.extremum_uniqueI
tff(fact_7461_ordering__top_Onot__eq__extremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( ( A3 != Top )
      <=> aa(A,$o,aa(A,fun(A,$o),Less,A3),Top) ) ) ).

% ordering_top.not_eq_extremum
tff(fact_7462_ordering__top_Oextremum__unique,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Top),A3)
      <=> ( A3 = Top ) ) ) ).

% ordering_top.extremum_unique
tff(fact_7463_ordering__top_Oextremum__strict,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => ~ aa(A,$o,aa(A,fun(A,$o),Less,Top),A3) ) ).

% ordering_top.extremum_strict
tff(fact_7464_ordering__top_Oextremum,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),Top: A,A3: A] :
      ( ordering_top(A,Less_eq,Less,Top)
     => aa(A,$o,aa(A,fun(A,$o),Less_eq,A3),Top) ) ).

% ordering_top.extremum
tff(fact_7465_admissible__disj,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),P)
         => ( comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),Q)
           => comple1908693960933563346ssible(A,complete_Sup_Sup(A),ord_less_eq(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aik(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) ) ) ) ).

% admissible_disj
tff(fact_7466_gcd__nat_Oordering__top__axioms,axiom,
    ordering_top(nat,dvd_dvd(nat),aTP_Lamp_ahn(nat,fun(nat,$o)),zero_zero(nat)) ).

% gcd_nat.ordering_top_axioms
tff(fact_7467_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_7468_bot__nat__0_Oordering__top__axioms,axiom,
    ordering_top(nat,aTP_Lamp_ag(nat,fun(nat,$o)),aTP_Lamp_af(nat,fun(nat,$o)),zero_zero(nat)) ).

% bot_nat_0.ordering_top_axioms
tff(fact_7469_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Na: nat] :
      ( aa(nat,$o,bit_se5641148757651400278ts_bit(int,bit_concat_bit(M,K,L)),Na)
    <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Na),M)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,K),Na) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na)
          & aa(nat,$o,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,minus_minus(nat,Na),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_7470_span__singleton,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A] : real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = aa(set(real),set(A),image2(real,A,aTP_Lamp_ail(A,fun(real,A),Xa)),top_top(set(real))) ) ).

% span_singleton
tff(fact_7471_concat__bit__0,axiom,
    ! [K: int,L: int] : bit_concat_bit(zero_zero(nat),K,L) = L ).

% concat_bit_0
tff(fact_7472_span__insert__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] : real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),S)) = real_Vector_span(A,S) ) ).

% span_insert_0
tff(fact_7473_concat__bit__nonnegative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_concat_bit(Na,K,L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_7474_concat__bit__negative__iff,axiom,
    ! [Na: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),bit_concat_bit(Na,K,L)),zero_zero(int))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_7475_span__empty,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ( real_Vector_span(A,bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))) ) ) ).

% span_empty
tff(fact_7476_span__delete__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] : real_Vector_span(A,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A))))) = real_Vector_span(A,S) ) ).

% span_delete_0
tff(fact_7477_dependent__insertD,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A,S: set(A)] :
          ( ~ member(A,A3,real_Vector_span(A,S))
         => ( real_V358717886546972837endent(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),S))
           => real_V358717886546972837endent(A,S) ) ) ) ).

% dependent_insertD
tff(fact_7478_independent__insert,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A,S: set(A)] :
          ( ~ real_V358717886546972837endent(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),S))
        <=> $ite(
              member(A,A3,S),
              ~ real_V358717886546972837endent(A,S),
              ( ~ real_V358717886546972837endent(A,S)
              & ~ member(A,A3,real_Vector_span(A,S)) ) ) ) ) ).

% independent_insert
tff(fact_7479_independent__insertI,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A,S: set(A)] :
          ( ~ member(A,A3,real_Vector_span(A,S))
         => ( ~ real_V358717886546972837endent(A,S)
           => ~ real_V358717886546972837endent(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),S)) ) ) ) ).

% independent_insertI
tff(fact_7480_span__redundant,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,S: set(A)] :
          ( member(A,Xa,real_Vector_span(A,S))
         => ( real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),S)) = real_Vector_span(A,S) ) ) ) ).

% span_redundant
tff(fact_7481_in__span__insert,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A,B3: A,S: set(A)] :
          ( member(A,A3,real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),S)))
         => ( ~ member(A,A3,real_Vector_span(A,S))
           => member(A,B3,real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),S))) ) ) ) ).

% in_span_insert
tff(fact_7482_span__trans,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,S: set(A),Ya: A] :
          ( member(A,Xa,real_Vector_span(A,S))
         => ( member(A,Ya,real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),S)))
           => member(A,Ya,real_Vector_span(A,S)) ) ) ) ).

% span_trans
tff(fact_7483_eq__span__insert__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,Ya: A,S: set(A)] :
          ( member(A,aa(A,A,minus_minus(A,Xa),Ya),real_Vector_span(A,S))
         => ( real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),S)) = real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),S)) ) ) ) ).

% eq_span_insert_eq
tff(fact_7484_in__span__delete,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A,S: set(A),B3: A] :
          ( member(A,A3,real_Vector_span(A,S))
         => ( ~ member(A,A3,real_Vector_span(A,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))))
           => member(A,B3,real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A))))))) ) ) ) ).

% in_span_delete
tff(fact_7485_span__breakdown__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,A3: A,S: set(A)] :
          ( member(A,Xa,real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),S)))
        <=> ? [K3: real] : member(A,aa(A,A,minus_minus(A,Xa),aa(A,A,real_V8093663219630862766scaleR(A,K3),A3)),real_Vector_span(A,S)) ) ) ).

% span_breakdown_eq
tff(fact_7486_span__induct__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xa: A,S: set(A),H: fun(A,$o)] :
          ( member(A,Xa,real_Vector_span(A,S))
         => ( aa(A,$o,H,zero_zero(A))
           => ( ! [C5: real,X3: A,Y: A] :
                  ( member(A,X3,S)
                 => ( aa(A,$o,H,Y)
                   => aa(A,$o,H,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,C5),X3)),Y)) ) )
             => aa(A,$o,H,Xa) ) ) ) ) ).

% span_induct_alt
tff(fact_7487_span__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] : member(A,zero_zero(A),real_Vector_span(A,S)) ) ).

% span_0
tff(fact_7488_span__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),T3: set(A)] :
          ( ( real_Vector_span(A,S) = real_Vector_span(A,T3) )
        <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),real_Vector_span(A,T3))
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),real_Vector_span(A,S)) ) ) ) ).

% span_eq
tff(fact_7489_span__mono,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),real_Vector_span(A,A4)),real_Vector_span(A,B2)) ) ) ).

% span_mono
tff(fact_7490_span__superset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),real_Vector_span(A,S)) ) ).

% span_superset
tff(fact_7491_maximal__independent__subset__extend,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),V: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),V)
         => ( ~ real_V358717886546972837endent(A,S)
           => ~ ! [B4: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),B4)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),V)
                   => ( ~ real_V358717886546972837endent(A,B4)
                     => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V),real_Vector_span(A,B4)) ) ) ) ) ) ) ).

% maximal_independent_subset_extend
tff(fact_7492_spanning__subset__independent,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A),A4: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
         => ( ~ real_V358717886546972837endent(A,A4)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),real_Vector_span(A,B2))
             => ( A4 = B2 ) ) ) ) ) ).

% spanning_subset_independent
tff(fact_7493_maximal__independent__subset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A)] :
          ~ ! [B4: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),V)
             => ( ~ real_V358717886546972837endent(A,B4)
               => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V),real_Vector_span(A,B4)) ) ) ) ).

% maximal_independent_subset
tff(fact_7494_span__Un,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),T3: set(A)] : real_Vector_span(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3)) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_aim(set(A),fun(set(A),fun(A,$o)),S),T3)) ) ).

% span_Un
tff(fact_7495_span__insert,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A,S: set(A)] : real_Vector_span(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),S)) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),aTP_Lamp_ain(A,fun(set(A),fun(A,$o)),A3),S)) ) ).

% span_insert
tff(fact_7496_span__finite,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( real_Vector_span(A,S) = aa(set(fun(A,real)),set(A),image2(fun(A,real),A,aTP_Lamp_aio(set(A),fun(fun(A,real),A),S)),top_top(set(fun(A,real)))) ) ) ) ).

% span_finite
tff(fact_7497_dependent__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [P: set(A)] :
          ( real_V358717886546972837endent(A,P)
        <=> ? [X: A] :
              ( member(A,X,P)
              & member(A,X,real_Vector_span(A,aa(set(A),set(A),minus_minus(set(A),P),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X),bot_bot(set(A)))))) ) ) ) ).

% dependent_def
tff(fact_7498_span__image__scale,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),C3: fun(A,real)] :
          ( aa(set(A),$o,finite_finite2(A),S)
         => ( ! [X3: A] :
                ( member(A,X3,S)
               => ( aa(A,real,C3,X3) != zero_zero(real) ) )
           => ( real_Vector_span(A,aa(set(A),set(A),image2(A,A,aTP_Lamp_aex(fun(A,real),fun(A,A),C3)),S)) = real_Vector_span(A,S) ) ) ) ) ).

% span_image_scale
tff(fact_7499_span__breakdown,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: A,S: set(A),A3: A] :
          ( member(A,B3,S)
         => ( member(A,A3,real_Vector_span(A,S))
           => ? [K2: real] : member(A,aa(A,A,minus_minus(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,K2),B3)),real_Vector_span(A,aa(set(A),set(A),minus_minus(set(A),S),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),bot_bot(set(A)))))) ) ) ) ).

% span_breakdown
tff(fact_7500_independent__span__bound,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [T3: set(A),S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( ~ real_V358717886546972837endent(A,S)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),real_Vector_span(A,T3))
             => ( aa(set(A),$o,finite_finite2(A),S)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(set(A),nat,finite_card(A),S)),aa(set(A),nat,finite_card(A),T3)) ) ) ) ) ) ).

% independent_span_bound
tff(fact_7501_exchange__lemma,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [T3: set(A),S: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),T3)
         => ( ~ real_V358717886546972837endent(A,S)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),real_Vector_span(A,T3))
             => ? [T12: set(A)] :
                  ( ( aa(set(A),nat,finite_card(A),T12) = aa(set(A),nat,finite_card(A),T3) )
                  & aa(set(A),$o,finite_finite2(A),T12)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T12)
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T12),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),S),T3))
                  & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),real_Vector_span(A,T12)) ) ) ) ) ) ).

% exchange_lemma
tff(fact_7502_span__alt,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] : real_Vector_span(A,B2) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aip(set(A),fun(A,$o),B2)) ) ).

% span_alt
tff(fact_7503_span__explicit_H,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] : real_Vector_span(A,B3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aiq(set(A),fun(A,$o),B3)) ) ).

% span_explicit'
tff(fact_7504_span__explicit,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B3: set(A)] : real_Vector_span(A,B3) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_air(set(A),fun(A,$o),B3)) ) ).

% span_explicit
tff(fact_7505_representation__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A] :
          real_V7696804695334737415tation(A,Basis,V2) = $ite(
            ( ~ real_V358717886546972837endent(A,Basis)
            & member(A,V2,real_Vector_span(A,Basis)) ),
            fChoice(fun(A,real),aa(A,fun(fun(A,real),$o),aTP_Lamp_ais(set(A),fun(A,fun(fun(A,real),$o)),Basis),V2)),
            aTP_Lamp_ait(A,real) ) ) ).

% representation_def
tff(fact_7506_extend__basis__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] : real_V4986007116245087402_basis(A,B2) = fChoice(set(A),aTP_Lamp_aiu(set(A),fun(set(A),$o),B2)) ) ).

% extend_basis_def
tff(fact_7507_representation__zero,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),X2: A] : aa(A,real,real_V7696804695334737415tation(A,Basis,zero_zero(A)),X2) = zero_zero(real) ) ).

% representation_zero
tff(fact_7508_finite__representation,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A] : aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_aiv(set(A),fun(A,fun(A,$o)),Basis),V2))) ) ).

% finite_representation
tff(fact_7509_representation__extend,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,Basis2: set(A)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( member(A,V2,real_Vector_span(A,Basis2))
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Basis2),Basis)
             => ( real_V7696804695334737415tation(A,Basis,V2) = real_V7696804695334737415tation(A,Basis2,V2) ) ) ) ) ) ).

% representation_extend
tff(fact_7510_extend__basis__superset,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A)] :
          ( ~ real_V358717886546972837endent(A,B2)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),real_V4986007116245087402_basis(A,B2)) ) ) ).

% extend_basis_superset
tff(fact_7511_sum__representation__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,B2: set(A)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( member(A,V2,real_Vector_span(A,Basis))
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Basis),B2)
               => ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aa(A,fun(A,A),aTP_Lamp_aiw(set(A),fun(A,fun(A,A)),Basis),V2)),B2) = V2 ) ) ) ) ) ) ).

% sum_representation_eq
tff(fact_7512_representation__eqI,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Basis: set(A),V2: A,F2: fun(A,real)] :
          ( ~ real_V358717886546972837endent(A,Basis)
         => ( member(A,V2,real_Vector_span(A,Basis))
           => ( ! [B5: A] :
                  ( ( aa(A,real,F2,B5) != zero_zero(real) )
                 => member(A,B5,Basis) )
             => ( aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F2)))
               => ( ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),F2)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F2))) = V2 )
                 => ( real_V7696804695334737415tation(A,Basis,V2) = F2 ) ) ) ) ) ) ) ).

% representation_eqI
tff(fact_7513_construct__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [B2: set(B),G: fun(B,A),V2: B] : real_V4425403222259421789struct(B,A,B2,G,V2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7311177749621191930dd_sum(B,A),aa(B,fun(B,A),aa(fun(B,A),fun(B,fun(B,A)),aTP_Lamp_aix(set(B),fun(fun(B,A),fun(B,fun(B,A))),B2),G),V2)),aa(fun(B,$o),set(B),collect(B),aa(B,fun(B,$o),aTP_Lamp_aiy(set(B),fun(B,fun(B,$o)),B2),V2))) ) ).

% construct_def
tff(fact_7514_dim__def,axiom,
    ! [V: set(a)] :
      real_Vector_dim(a,V) = $ite(
        ? [B7: set(a)] :
          ( ~ real_V358717886546972837endent(a,B7)
          & ( real_Vector_span(a,B7) = real_Vector_span(a,V) ) ),
        aa(set(a),nat,finite_card(a),fChoice(set(a),aTP_Lamp_aiz(set(a),fun(set(a),$o),V))),
        zero_zero(nat) ) ).

% dim_def
tff(fact_7515_dim__le__card_H,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S3: set(A)] :
          ( aa(set(A),$o,finite_finite2(A),S3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),real_Vector_dim(A,S3)),aa(set(A),nat,finite_card(A),S3)) ) ) ).

% dim_le_card'
tff(fact_7516_dim__unique,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A),V: set(A),Na: nat] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),V)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V),real_Vector_span(A,B2))
           => ( ~ real_V358717886546972837endent(A,B2)
             => ( ( aa(set(A),nat,finite_card(A),B2) = Na )
               => ( real_Vector_dim(A,V) = Na ) ) ) ) ) ) ).

% dim_unique
tff(fact_7517_basis__exists,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A)] :
          ~ ! [B4: set(A)] :
              ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B4),V)
             => ( ~ real_V358717886546972837endent(A,B4)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V),real_Vector_span(A,B4))
                 => ( aa(set(A),nat,finite_card(A),B4) != real_Vector_dim(A,V) ) ) ) ) ) ).

% basis_exists
tff(fact_7518_basis__card__eq__dim,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A),V: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),V)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V),real_Vector_span(A,B2))
           => ( ~ real_V358717886546972837endent(A,B2)
             => ( aa(set(A),nat,finite_card(A),B2) = real_Vector_dim(A,V) ) ) ) ) ) ).

% basis_card_eq_dim
tff(fact_7519_dim__le__card,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [V: set(A),W3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V),real_Vector_span(A,W3))
         => ( aa(set(A),$o,finite_finite2(A),W3)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),real_Vector_dim(A,V)),aa(set(A),nat,finite_card(A),W3)) ) ) ) ).

% dim_le_card
tff(fact_7520_span__card__ge__dim,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: set(A),V: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),V)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V),real_Vector_span(A,B2))
           => ( aa(set(A),$o,finite_finite2(A),B2)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),real_Vector_dim(A,V)),aa(set(A),nat,finite_card(A),B2)) ) ) ) ) ).

% span_card_ge_dim
tff(fact_7521_construct__outside,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [B2: set(A),V2: A,F2: fun(A,B)] :
          ( ~ real_V358717886546972837endent(A,B2)
         => ( member(A,V2,real_Vector_span(A,aa(set(A),set(A),minus_minus(set(A),real_V4986007116245087402_basis(A,B2)),B2)))
           => ( real_V4425403222259421789struct(A,B,B2,F2,V2) = zero_zero(B) ) ) ) ) ).

% construct_outside
tff(fact_7522_linear__indep__image__lemma,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),B2: set(A),Xa: A] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ~ real_V358717886546972837endent(B,aa(set(A),set(B),image2(A,B,F2),B2))
             => ( inj_on(A,B,F2,B2)
               => ( member(A,Xa,real_Vector_span(A,B2))
                 => ( ( aa(A,B,F2,Xa) = zero_zero(B) )
                   => ( Xa = zero_zero(A) ) ) ) ) ) ) ) ) ).

% linear_indep_image_lemma
tff(fact_7523_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,$o),Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs)))
     => ( aa(nat,A,nth(A,dropWhile(A,P,Xs)),J) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% dropWhile_nth
tff(fact_7524_linear__eq__0__on__span,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B),B3: set(A),Xa: A] :
          ( real_Vector_linear(A,B,F2)
         => ( ! [X3: A] :
                ( member(A,X3,B3)
               => ( aa(A,B,F2,X3) = zero_zero(B) ) )
           => ( member(A,Xa,real_Vector_span(A,B3))
             => ( aa(A,B,F2,Xa) = zero_zero(B) ) ) ) ) ) ).

% linear_eq_0_on_span
tff(fact_7525_length__dropWhile__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_dropWhile_le
tff(fact_7526_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_7527_module__hom__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => real_Vector_linear(A,B,aTP_Lamp_aja(A,B)) ) ).

% module_hom_zero
tff(fact_7528_linear__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(A,B,F2,zero_zero(A)) = zero_zero(B) ) ) ) ).

% linear_0
tff(fact_7529_linear__injective__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => ( inj_on(A,B,F2,top_top(set(A)))
          <=> ! [X: A] :
                ( ( aa(A,B,F2,X) = zero_zero(B) )
               => ( X = zero_zero(A) ) ) ) ) ) ).

% linear_injective_0
tff(fact_7530_linear__spans__image,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),V: set(A),B2: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),V),real_Vector_span(A,B2))
           => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),V)),real_Vector_span(B,aa(set(A),set(B),image2(A,B,F2),B2))) ) ) ) ).

% linear_spans_image
tff(fact_7531_linear__surj__right__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B),T3: set(B),S: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),real_Vector_span(B,T3)),aa(set(A),set(B),image2(A,B,F2),real_Vector_span(A,S)))
           => ? [G7: fun(B,A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G7),top_top(set(B)))),real_Vector_span(A,S))
                & real_Vector_linear(B,A,G7)
                & ! [X2: B] :
                    ( member(B,X2,real_Vector_span(B,T3))
                   => ( aa(A,B,F2,aa(B,A,G7,X2)) = X2 ) ) ) ) ) ) ).

% linear_surj_right_inverse
tff(fact_7532_linear__spanning__surjective__image,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),real_Vector_span(A,S))
           => ( ( aa(set(A),set(B),image2(A,B,F2),top_top(set(A))) = top_top(set(B)) )
             => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),top_top(set(B))),real_Vector_span(B,aa(set(A),set(B),image2(A,B,F2),S))) ) ) ) ) ).

% linear_spanning_surjective_image
tff(fact_7533_linear__inj__on__left__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),S: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( inj_on(A,B,F2,real_Vector_span(A,S))
           => ? [G7: fun(B,A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G7),top_top(set(B)))),real_Vector_span(A,S))
                & real_Vector_linear(B,A,G7)
                & ! [X2: A] :
                    ( member(A,X2,real_Vector_span(A,S))
                   => ( aa(B,A,G7,aa(A,B,F2,X2)) = X2 ) ) ) ) ) ) ).

% linear_inj_on_left_inverse
tff(fact_7534_finite__basis__to__basis__subspace__isomorphism,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [S: set(A),T3: set(B),B2: set(A),C2: set(B)] :
          ( real_Vector_subspace(A,S)
         => ( real_Vector_subspace(B,T3)
           => ( ( real_Vector_dim(A,S) = real_Vector_dim(B,T3) )
             => ( aa(set(A),$o,finite_finite2(A),B2)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),S)
                 => ( ~ real_V358717886546972837endent(A,B2)
                   => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),real_Vector_span(A,B2))
                     => ( ( aa(set(A),nat,finite_card(A),B2) = real_Vector_dim(A,S) )
                       => ( aa(set(B),$o,finite_finite2(B),C2)
                         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C2),T3)
                           => ( ~ real_V358717886546972837endent(B,C2)
                             => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T3),real_Vector_span(B,C2))
                               => ( ( aa(set(B),nat,finite_card(B),C2) = real_Vector_dim(B,T3) )
                                 => ? [F5: fun(A,B)] :
                                      ( real_Vector_linear(A,B,F5)
                                      & ( aa(set(A),set(B),image2(A,B,F5),B2) = C2 )
                                      & ( aa(set(A),set(B),image2(A,B,F5),S) = T3 )
                                      & inj_on(A,B,F5,S) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% finite_basis_to_basis_subspace_isomorphism
tff(fact_7535_linear__exists__left__inverse__on,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),V: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( real_Vector_subspace(A,V)
           => ( inj_on(A,B,F2,V)
             => ? [G7: fun(B,A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G7),top_top(set(B)))),V)
                  & real_Vector_linear(B,A,G7)
                  & ! [X2: A] :
                      ( member(A,X2,V)
                     => ( aa(B,A,G7,aa(A,B,F2,X2)) = X2 ) ) ) ) ) ) ) ).

% linear_exists_left_inverse_on
tff(fact_7536_linear__subspace__kernel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B)] :
          ( real_Vector_linear(A,B,F2)
         => real_Vector_subspace(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ajb(fun(A,B),fun(A,$o),F2))) ) ) ).

% linear_subspace_kernel
tff(fact_7537_subspace__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( real_Vector_subspace(A,S)
         => member(A,zero_zero(A),S) ) ) ).

% subspace_0
tff(fact_7538_span__subspace,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A4: set(A),B2: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),real_Vector_span(A,A4))
           => ( real_Vector_subspace(A,B2)
             => ( real_Vector_span(A,A4) = B2 ) ) ) ) ) ).

% span_subspace
tff(fact_7539_span__minimal,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),T3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( real_Vector_subspace(A,T3)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),real_Vector_span(A,S)),T3) ) ) ) ).

% span_minimal
tff(fact_7540_span__unique,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A),T3: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T3)
         => ( real_Vector_subspace(A,T3)
           => ( ! [T13: set(A)] :
                  ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),T13)
                 => ( real_Vector_subspace(A,T13)
                   => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T3),T13) ) )
             => ( real_Vector_span(A,S) = T3 ) ) ) ) ) ).

% span_unique
tff(fact_7541_subspace__single__0,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => real_Vector_subspace(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),zero_zero(A)),bot_bot(set(A)))) ) ).

% subspace_single_0
tff(fact_7542_subspace__def,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( real_Vector_subspace(A,S)
        <=> ( member(A,zero_zero(A),S)
            & ! [X: A] :
                ( member(A,X,S)
               => ! [Xa3: A] :
                    ( member(A,Xa3,S)
                   => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Xa3),S) ) )
            & ! [C6: real,X: A] :
                ( member(A,X,S)
               => member(A,aa(A,A,real_V8093663219630862766scaleR(A,C6),X),S) ) ) ) ) ).

% subspace_def
tff(fact_7543_subspaceI,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [S: set(A)] :
          ( member(A,zero_zero(A),S)
         => ( ! [X3: A,Y: A] :
                ( member(A,X3,S)
               => ( member(A,Y,S)
                 => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y),S) ) )
           => ( ! [C5: real,X3: A] :
                  ( member(A,X3,S)
                 => member(A,aa(A,A,real_V8093663219630862766scaleR(A,C5),X3),S) )
             => real_Vector_subspace(A,S) ) ) ) ) ).

% subspaceI
tff(fact_7544_linear__injective__on__subspace__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [F2: fun(A,B),S3: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( real_Vector_subspace(A,S3)
           => ( inj_on(A,B,F2,S3)
            <=> ! [X: A] :
                  ( member(A,X,S3)
                 => ( ( aa(A,B,F2,X) = zero_zero(B) )
                   => ( X = zero_zero(A) ) ) ) ) ) ) ) ).

% linear_injective_on_subspace_0
tff(fact_7545_linear__exists__right__inverse__on,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [F2: fun(A,B),V: set(A)] :
          ( real_Vector_linear(A,B,F2)
         => ( real_Vector_subspace(A,V)
           => ? [G7: fun(B,A)] :
                ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image2(B,A,G7),top_top(set(B)))),V)
                & real_Vector_linear(B,A,G7)
                & ! [X2: B] :
                    ( member(B,X2,aa(set(A),set(B),image2(A,B,F2),V))
                   => ( aa(A,B,F2,aa(B,A,G7,X2)) = X2 ) ) ) ) ) ) ).

% linear_exists_right_inverse_on
tff(fact_7546_less__eq__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_acp(nat,fun(nat,fun(product_prod(nat,nat),$o))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_acp(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_eq_int.rsp
tff(fact_7547_less__int_Orsp,axiom,
    aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o,aa(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),fun(fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),$o),bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),fun(product_prod(nat,nat),$o),fun(product_prod(nat,nat),$o),intrel,bNF_rel_fun(product_prod(nat,nat),product_prod(nat,nat),$o,$o,intrel,fequal($o))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_act(nat,fun(nat,fun(product_prod(nat,nat),$o))))),aa(fun(nat,fun(nat,fun(product_prod(nat,nat),$o))),fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)),product_case_prod(nat,nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_act(nat,fun(nat,fun(product_prod(nat,nat),$o))))) ).

% less_int.rsp
tff(fact_7548_zero__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),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_7549_one__int_Orsp,axiom,
    aa(product_prod(nat,nat),$o,aa(product_prod(nat,nat),fun(product_prod(nat,nat),$o),intrel,aa(nat,product_prod(nat,nat),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_7550_euclidean__size__times__nonunit,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,B3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))) ) ) ) ) ).

% euclidean_size_times_nonunit
tff(fact_7551_less__eq__enat__def,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),M),Na)
    <=> extended_case_enat($o,aTP_Lamp_ajc(extended_enat,fun(nat,$o),M),$true,Na) ) ).

% less_eq_enat_def
tff(fact_7552_size__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ( euclid6346220572633701492n_size(A,zero_zero(A)) = zero_zero(nat) ) ) ).

% size_0
tff(fact_7553_euclidean__size__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A] :
          ( ( euclid6346220572633701492n_size(A,B3) = zero_zero(nat) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% euclidean_size_eq_0_iff
tff(fact_7554_euclidean__size__greater__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),euclid6346220572633701492n_size(A,B3))
        <=> ( B3 != zero_zero(A) ) ) ) ).

% euclidean_size_greater_0_iff
tff(fact_7555_dvd__euclidean__size__eq__imp__dvd,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,B3) )
           => ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),A3)
             => aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3) ) ) ) ) ).

% dvd_euclidean_size_eq_imp_dvd
tff(fact_7556_unit__iff__euclidean__size,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),one_one(A))
        <=> ( ( euclid6346220572633701492n_size(A,A3) = euclid6346220572633701492n_size(A,one_one(A)) )
            & ( A3 != zero_zero(A) ) ) ) ) ).

% unit_iff_euclidean_size
tff(fact_7557_size__mult__mono,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))) ) ) ).

% size_mult_mono
tff(fact_7558_size__mult__mono_H,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3))) ) ) ).

% size_mult_mono'
tff(fact_7559_dvd__proper__imp__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3)
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),A3)
           => ( ( B3 != zero_zero(A) )
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B3)) ) ) ) ) ).

% dvd_proper_imp_size_less
tff(fact_7560_dvd__imp__size__le,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),A3),B3)
         => ( ( B3 != zero_zero(A) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B3)) ) ) ) ).

% dvd_imp_size_le
tff(fact_7561_mod__size__less,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,modulo_modulo(A,A3,B3))),euclid6346220572633701492n_size(A,B3)) ) ) ).

% mod_size_less
tff(fact_7562_less__enat__def,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M),Na)
    <=> extended_case_enat($o,aTP_Lamp_ajd(extended_enat,fun(nat,$o),Na),$false,M) ) ).

% less_enat_def
tff(fact_7563_divmod__cases,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,A3: A] :
          ( ( ( B3 != zero_zero(A) )
           => ( ( modulo_modulo(A,A3,B3) = zero_zero(A) )
             => ( A3 != aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A3,B3)),B3) ) ) )
         => ( ( ( B3 != zero_zero(A) )
             => ! [Q3: A,R3: A] :
                  ( ( euclid7384307370059645450egment(A,R3) = euclid7384307370059645450egment(A,B3) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R3)),euclid6346220572633701492n_size(A,B3))
                   => ( ( R3 != zero_zero(A) )
                     => ( ( divide_divide(A,A3,B3) = Q3 )
                       => ( ( modulo_modulo(A,A3,B3) = R3 )
                         => ( A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q3),B3)),R3) ) ) ) ) ) ) )
           => ( B3 = zero_zero(A) ) ) ) ) ).

% divmod_cases
tff(fact_7564_mod__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,R2: A,Q5: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B3)),R2) = A3 )
               => ( modulo_modulo(A,A3,B3) = R2 ) ) ) ) ) ) ).

% mod_eqI
tff(fact_7565_division__segment__not__0,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A] : euclid7384307370059645450egment(A,A3) != zero_zero(A) ) ).

% division_segment_not_0
tff(fact_7566_division__segment__mult,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( euclid7384307370059645450egment(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),euclid7384307370059645450egment(A,A3)),euclid7384307370059645450egment(A,B3)) ) ) ) ) ).

% division_segment_mult
tff(fact_7567_division__segment__mod,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ~ aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),B3),A3)
           => ( euclid7384307370059645450egment(A,modulo_modulo(A,A3,B3)) = euclid7384307370059645450egment(A,B3) ) ) ) ) ).

% division_segment_mod
tff(fact_7568_division__segment__int__def,axiom,
    ! [K: int] :
      euclid7384307370059645450egment(int,K) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ).

% division_segment_int_def
tff(fact_7569_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B3: A] :
          ( ( euclid7384307370059645450egment(A,A3) = euclid7384307370059645450egment(A,B3) )
         => ( ( divide_divide(A,A3,B3) = zero_zero(A) )
          <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,A3)),euclid6346220572633701492n_size(A,B3))
              | ( B3 = zero_zero(A) ) ) ) ) ) ).

% unique_euclidean_semiring_class.div_eq_0_iff
tff(fact_7570_div__bounded,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,R2: A,Q5: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
             => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B3)),R2),B3) = Q5 ) ) ) ) ) ).

% div_bounded
tff(fact_7571_div__eqI,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [B3: A,R2: A,Q5: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( euclid7384307370059645450egment(A,R2) = euclid7384307370059645450egment(A,B3) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),euclid6346220572633701492n_size(A,R2)),euclid6346220572633701492n_size(A,B3))
             => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Q5),B3)),R2) = A3 )
               => ( divide_divide(A,A3,B3) = Q5 ) ) ) ) ) ) ).

% div_eqI
tff(fact_7572_sorted__wrt__iff__nth__Suc__transp,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o)),Xs: list(A)] :
      ( transp(A,P)
     => ( sorted_wrt(A,P,Xs)
      <=> ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,suc,I4)),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(A,fun(A,$o),P,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Xs),aa(nat,nat,suc,I4))) ) ) ) ).

% sorted_wrt_iff_nth_Suc_transp
tff(fact_7573_total__inv__image,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),R2: set(product_prod(B,B))] :
      ( inj_on(A,B,F2,top_top(set(A)))
     => ( total_on(B,top_top(set(B)),R2)
       => total_on(A,top_top(set(A)),inv_image(B,A,R2,F2)) ) ) ).

% total_inv_image
tff(fact_7574_in__inv__image,axiom,
    ! [A: $tType,B: $tType,Xa: A,Ya: A,R2: set(product_prod(B,B)),F2: fun(A,B)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),inv_image(B,A,R2,F2))
    <=> member(product_prod(B,B),aa(B,product_prod(B,B),product_Pair(B,B,aa(A,B,F2,Xa)),aa(A,B,F2,Ya)),R2) ) ).

% in_inv_image
tff(fact_7575_converse__inv__image,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(B,B)),F2: fun(A,B)] : converse(A,A,inv_image(B,A,R,F2)) = inv_image(B,A,converse(B,B,R),F2) ).

% converse_inv_image
tff(fact_7576_transp__INF,axiom,
    ! [B: $tType,A: $tType,S: set(A),R2: fun(A,fun(B,fun(B,$o)))] :
      ( ! [X3: A] :
          ( member(A,X3,S)
         => transp(B,aa(A,fun(B,fun(B,$o)),R2,X3)) )
     => transp(B,aa(set(fun(B,fun(B,$o))),fun(B,fun(B,$o)),complete_Inf_Inf(fun(B,fun(B,$o))),aa(set(A),set(fun(B,fun(B,$o))),image2(A,fun(B,fun(B,$o)),R2),S))) ) ).

% transp_INF
tff(fact_7577_transp__trans__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( transp(A,aTP_Lamp_ahx(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> trans(A,R2) ) ).

% transp_trans_eq
tff(fact_7578_transp__trans,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( transp(A,R2)
    <=> trans(A,aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),R2))) ) ).

% transp_trans
tff(fact_7579_trans__inv__image,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),F2: fun(B,A)] :
      ( trans(A,R2)
     => trans(B,inv_image(A,B,R2,F2)) ) ).

% trans_inv_image
tff(fact_7580_transp__inf,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),S3: fun(A,fun(A,$o))] :
      ( transp(A,R2)
     => ( transp(A,S3)
       => transp(A,aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),inf_inf(fun(A,fun(A,$o))),R2),S3)) ) ) ).

% transp_inf
tff(fact_7581_transp__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less(A)) ) ).

% transp_less
tff(fact_7582_transpD,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: A,Ya: A,Z: A] :
      ( transp(A,R2)
     => ( aa(A,$o,aa(A,fun(A,$o),R2,Xa),Ya)
       => ( aa(A,$o,aa(A,fun(A,$o),R2,Ya),Z)
         => aa(A,$o,aa(A,fun(A,$o),R2,Xa),Z) ) ) ) ).

% transpD
tff(fact_7583_transpE,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),Xa: A,Ya: A,Z: A] :
      ( transp(A,R2)
     => ( aa(A,$o,aa(A,fun(A,$o),R2,Xa),Ya)
       => ( aa(A,$o,aa(A,fun(A,$o),R2,Ya),Z)
         => aa(A,$o,aa(A,fun(A,$o),R2,Xa),Z) ) ) ) ).

% transpE
tff(fact_7584_transpI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( ! [X3: A,Y: A,Z2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,X3),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),R2,Y),Z2)
           => aa(A,$o,aa(A,fun(A,$o),R2,X3),Z2) ) )
     => transp(A,R2) ) ).

% transpI
tff(fact_7585_transp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( transp(A,R2)
    <=> ! [X: A,Y4: A,Z5: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,X),Y4)
         => ( aa(A,$o,aa(A,fun(A,$o),R2,Y4),Z5)
           => aa(A,$o,aa(A,fun(A,$o),R2,X),Z5) ) ) ) ).

% transp_def
tff(fact_7586_transp__equality,axiom,
    ! [A: $tType] : transp(A,fequal(A)) ).

% transp_equality
tff(fact_7587_transp__empty,axiom,
    ! [A: $tType] : transp(A,aTP_Lamp_aje(A,fun(A,$o))) ).

% transp_empty
tff(fact_7588_transp__singleton,axiom,
    ! [A: $tType,A3: A] : transp(A,aTP_Lamp_ajf(A,fun(A,fun(A,$o)),A3)) ).

% transp_singleton
tff(fact_7589_transp__gr,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_ahy(A,fun(A,$o))) ) ).

% transp_gr
tff(fact_7590_transp__ge,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,aTP_Lamp_ajg(A,fun(A,$o))) ) ).

% transp_ge
tff(fact_7591_transp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => transp(A,ord_less_eq(A)) ) ).

% transp_le
tff(fact_7592_inv__image__def,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(B,B)),F2: fun(A,B)] : inv_image(B,A,R2,F2) = aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_ajh(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),R2),F2))) ).

% inv_image_def
tff(fact_7593_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),field_char_0_of_rat(A,R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),one_one(rat)) ) ) ).

% of_rat_le_1_iff
tff(fact_7594_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),field_char_0_of_rat(A,R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),one_one(rat)),R2) ) ) ).

% one_le_of_rat_iff
tff(fact_7595_zero__eq__of__rat__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( zero_zero(A) = field_char_0_of_rat(A,A3) )
        <=> ( zero_zero(rat) = A3 ) ) ) ).

% zero_eq_of_rat_iff
tff(fact_7596_of__rat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: rat] :
          ( ( field_char_0_of_rat(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(rat) ) ) ) ).

% of_rat_eq_0_iff
tff(fact_7597_of__rat__0,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_of_rat(A,zero_zero(rat)) = zero_zero(A) ) ) ).

% of_rat_0
tff(fact_7598_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),field_char_0_of_rat(A,R2)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),zero_zero(rat)) ) ) ).

% of_rat_less_0_iff
tff(fact_7599_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),zero_zero(A)),field_char_0_of_rat(A,R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R2) ) ) ).

% zero_less_of_rat_iff
tff(fact_7600_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),one_one(A)),field_char_0_of_rat(A,R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),one_one(rat)),R2) ) ) ).

% one_less_of_rat_iff
tff(fact_7601_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),field_char_0_of_rat(A,R2)),one_one(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),one_one(rat)) ) ) ).

% of_rat_less_1_iff
tff(fact_7602_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),field_char_0_of_rat(A,R2))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),zero_zero(rat)),R2) ) ) ).

% zero_le_of_rat_iff
tff(fact_7603_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),field_char_0_of_rat(A,R2)),zero_zero(A))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),zero_zero(rat)) ) ) ).

% of_rat_le_0_iff
tff(fact_7604_of__rat__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat,S3: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),field_char_0_of_rat(A,R2)),field_char_0_of_rat(A,S3))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),R2),S3) ) ) ).

% of_rat_less_eq
tff(fact_7605_of__rat__dense,axiom,
    ! [Xa: real,Ya: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),Ya)
     => ? [Q3: rat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),field_char_0_of_rat(real,Q3))
          & aa(real,$o,aa(real,fun(real,$o),ord_less(real),field_char_0_of_rat(real,Q3)),Ya) ) ) ).

% of_rat_dense
tff(fact_7606_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R2: rat,S3: rat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less(A),field_char_0_of_rat(A,R2)),field_char_0_of_rat(A,S3))
        <=> aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R2),S3) ) ) ).

% of_rat_less
tff(fact_7607_less__RealD,axiom,
    ! [Y3: fun(nat,rat),Xa: real] :
      ( cauchy(Y3)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),real2(Y3))
       => ? [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less(real),Xa),field_char_0_of_rat(real,aa(nat,rat,Y3,N2))) ) ) ).

% less_RealD
tff(fact_7608_le__RealI,axiom,
    ! [Y3: fun(nat,rat),Xa: real] :
      ( cauchy(Y3)
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),field_char_0_of_rat(real,aa(nat,rat,Y3,N2)))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xa),real2(Y3)) ) ) ).

% le_RealI
tff(fact_7609_Real__leI,axiom,
    ! [X4: fun(nat,rat),Ya: real] :
      ( cauchy(X4)
     => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),field_char_0_of_rat(real,aa(nat,rat,X4,N2))),Ya)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real2(X4)),Ya) ) ) ).

% Real_leI
tff(fact_7610_sym__trans__comp__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( sym(A,R2)
     => ( trans(A,R2)
       => aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,converse(A,A,R2),R2)),R2) ) ) ).

% sym_trans_comp_subset
tff(fact_7611_bounded__bilinear__def,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
        <=> ( ! [A9: A,A21: A,B7: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A9),A21)),B7) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A9),B7)),aa(B,C,aa(A,fun(B,C),Prod,A21),B7))
            & ! [A9: A,B7: B,B15: B] : aa(B,C,aa(A,fun(B,C),Prod,A9),aa(B,B,aa(B,fun(B,B),plus_plus(B),B7),B15)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A9),B7)),aa(B,C,aa(A,fun(B,C),Prod,A9),B15))
            & ! [R5: real,A9: A,B7: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,real_V8093663219630862766scaleR(A,R5),A9)),B7) = aa(C,C,real_V8093663219630862766scaleR(C,R5),aa(B,C,aa(A,fun(B,C),Prod,A9),B7))
            & ! [A9: A,R5: real,B7: B] : aa(B,C,aa(A,fun(B,C),Prod,A9),aa(B,B,real_V8093663219630862766scaleR(B,R5),B7)) = aa(C,C,real_V8093663219630862766scaleR(C,R5),aa(B,C,aa(A,fun(B,C),Prod,A9),B7))
            & ? [K5: real] :
              ! [A9: A,B7: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A9),B7))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A9)),real_V7770717601297561774m_norm(B,B7))),K5)) ) ) ) ).

% bounded_bilinear_def
tff(fact_7612_sym__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( sym(A,converse(A,A,R2))
    <=> sym(A,R2) ) ).

% sym_converse
tff(fact_7613_sym__UNION,axiom,
    ! [B: $tType,A: $tType,S: set(A),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [X3: A] :
          ( member(A,X3,S)
         => sym(B,aa(A,set(product_prod(B,B)),R2,X3)) )
     => sym(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).

% sym_UNION
tff(fact_7614_symD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: A,A3: A] :
      ( sym(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),A3),R2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2) ) ) ).

% symD
tff(fact_7615_symE,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B3: A,A3: A] :
      ( sym(A,R2)
     => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),A3),R2)
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),B3),R2) ) ) ).

% symE
tff(fact_7616_symI,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ! [A5: A,B5: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A5),B5),R2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B5),A5),R2) )
     => sym(A,R2) ) ).

% symI
tff(fact_7617_sym__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( sym(A,R2)
    <=> ! [X: A,Y4: A] :
          ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4),R2)
         => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Y4),X),R2) ) ) ).

% sym_def
tff(fact_7618_sym__Un,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( sym(A,R2)
     => ( sym(A,S3)
       => sym(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))),R2),S3)) ) ) ).

% sym_Un
tff(fact_7619_sym__Int,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( sym(A,R2)
     => ( sym(A,S3)
       => sym(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),S3)) ) ) ).

% sym_Int
tff(fact_7620_sym__Id__on,axiom,
    ! [A: $tType,A4: set(A)] : sym(A,id_on(A,A4)) ).

% sym_Id_on
tff(fact_7621_sym__Id,axiom,
    ! [A: $tType] : sym(A,id2(A)) ).

% sym_Id
tff(fact_7622_bounded__bilinear_Ozero__right,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),A3: A] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,A3),zero_zero(B)) = zero_zero(C) ) ) ) ).

% bounded_bilinear.zero_right
tff(fact_7623_bounded__bilinear_Ozero__left,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Prod: fun(A,fun(B,C)),B3: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( aa(B,C,aa(A,fun(B,C),Prod,zero_zero(A)),B3) = zero_zero(C) ) ) ) ).

% bounded_bilinear.zero_left
tff(fact_7624_sym__conv__converse__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( sym(A,R2)
    <=> ( converse(A,A,R2) = R2 ) ) ).

% sym_conv_converse_eq
tff(fact_7625_sym__Int__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : sym(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),converse(A,A,R2))) ).

% sym_Int_converse
tff(fact_7626_sym__Un__converse,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : sym(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))),R2),converse(A,A,R2))) ).

% sym_Un_converse
tff(fact_7627_sym__inv__image,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),F2: fun(B,A)] :
      ( sym(A,R2)
     => sym(B,inv_image(A,B,R2,F2)) ) ).

% sym_inv_image
tff(fact_7628_bounded__bilinear_Obounded,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ? [K8: real] :
            ! [A10: A,B10: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A10),B10))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A10)),real_V7770717601297561774m_norm(B,B10))),K8)) ) ) ).

% bounded_bilinear.bounded
tff(fact_7629_sym__INTER,axiom,
    ! [B: $tType,A: $tType,S: set(A),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [X3: A] :
          ( member(A,X3,S)
         => sym(B,aa(A,set(product_prod(B,B)),R2,X3)) )
     => sym(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Inf_Inf(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),S))) ) ).

% sym_INTER
tff(fact_7630_bounded__bilinear_Otendsto__right__zero,axiom,
    ! [C: $tType,B: $tType,A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,B),F3: filter(D),C3: A] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,B,F2,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
           => filterlim(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aji(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Prod),F2),C3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ).

% bounded_bilinear.tendsto_right_zero
tff(fact_7631_bounded__bilinear_Otendsto__left__zero,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,A),F3: filter(D),C3: B] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => filterlim(D,C,aa(B,fun(D,C),aa(fun(D,A),fun(B,fun(D,C)),aTP_Lamp_ajj(fun(A,fun(B,C)),fun(fun(D,A),fun(B,fun(D,C))),Prod),F2),C3),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ).

% bounded_bilinear.tendsto_left_zero
tff(fact_7632_bounded__bilinear_Otendsto__zero,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C)),F2: fun(D,A),F3: filter(D),G: fun(D,B)] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ( filterlim(D,A,F2,topolo7230453075368039082e_nhds(A,zero_zero(A)),F3)
           => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F3)
             => filterlim(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_ajk(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Prod),F2),G),topolo7230453075368039082e_nhds(C,zero_zero(C)),F3) ) ) ) ) ).

% bounded_bilinear.tendsto_zero
tff(fact_7633_bounded__bilinear_Ononneg__bounded,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),K8)
              & ! [A10: A,B10: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A10),B10))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A10)),real_V7770717601297561774m_norm(B,B10))),K8)) ) ) ) ).

% bounded_bilinear.nonneg_bounded
tff(fact_7634_bounded__bilinear_Opos__bounded,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( real_V2442710119149674383linear(A,B,C,Prod)
         => ? [K8: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),K8)
              & ! [A10: A,B10: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A10),B10))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A10)),real_V7770717601297561774m_norm(B,B10))),K8)) ) ) ) ).

% bounded_bilinear.pos_bounded
tff(fact_7635_bounded__bilinear_Ointro,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Prod: fun(A,fun(B,C))] :
          ( ! [A5: A,A20: A,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),A20)),B5) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A5),B5)),aa(B,C,aa(A,fun(B,C),Prod,A20),B5))
         => ( ! [A5: A,B5: B,B14: B] : aa(B,C,aa(A,fun(B,C),Prod,A5),aa(B,B,aa(B,fun(B,B),plus_plus(B),B5),B14)) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(B,C,aa(A,fun(B,C),Prod,A5),B5)),aa(B,C,aa(A,fun(B,C),Prod,A5),B14))
           => ( ! [R3: real,A5: A,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,aa(A,A,real_V8093663219630862766scaleR(A,R3),A5)),B5) = aa(C,C,real_V8093663219630862766scaleR(C,R3),aa(B,C,aa(A,fun(B,C),Prod,A5),B5))
             => ( ! [A5: A,R3: real,B5: B] : aa(B,C,aa(A,fun(B,C),Prod,A5),aa(B,B,real_V8093663219630862766scaleR(B,R3),B5)) = aa(C,C,real_V8093663219630862766scaleR(C,R3),aa(B,C,aa(A,fun(B,C),Prod,A5),B5))
               => ( ? [K6: real] :
                    ! [A5: A,B5: B] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(B,C,aa(A,fun(B,C),Prod,A5),B5))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A5)),real_V7770717601297561774m_norm(B,B5))),K6))
                 => real_V2442710119149674383linear(A,B,C,Prod) ) ) ) ) ) ) ).

% bounded_bilinear.intro
tff(fact_7636_coinduct3__lemma,axiom,
    ! [A: $tType,X4: set(A),F2: fun(set(A),set(A))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(A),set(A),F2,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_ajl(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X4),F2))))
     => ( aa(fun(set(A),set(A)),$o,order_mono(set(A),set(A)),F2)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_ajl(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X4),F2))),aa(set(A),set(A),F2,complete_lattice_lfp(set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_ajl(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),X4),F2)))) ) ) ).

% coinduct3_lemma
tff(fact_7637_def__coinduct3,axiom,
    ! [A: $tType,A4: set(A),F2: fun(set(A),set(A)),A3: A,X4: set(A)] :
      ( ( A4 = complete_lattice_gfp(set(A),F2) )
     => ( aa(fun(set(A),set(A)),$o,order_mono(set(A),set(A)),F2)
       => ( member(A,A3,X4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(A),set(A),F2,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_ajm(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),A4),F2),X4))))
           => member(A,A3,A4) ) ) ) ) ).

% def_coinduct3
tff(fact_7638_gfp__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,fun(A,A))] :
          ( ! [X3: A,Y: A,W: A,Z2: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F2,X3),W)),aa(A,A,aa(A,fun(A,A),F2,Y),Z2)) ) )
         => ( complete_lattice_gfp(A,aTP_Lamp_ajn(fun(A,fun(A,A)),fun(A,A),F2)) = complete_lattice_gfp(A,aTP_Lamp_aer(fun(A,fun(A,A)),fun(A,A),F2)) ) ) ) ).

% gfp_gfp
tff(fact_7639_weak__coinduct,axiom,
    ! [A: $tType,A3: A,X4: set(A),F2: fun(set(A),set(A))] :
      ( member(A,A3,X4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(A),set(A),F2,X4))
       => member(A,A3,complete_lattice_gfp(set(A),F2)) ) ) ).

% weak_coinduct
tff(fact_7640_gfp__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),G: fun(A,A)] :
          ( ! [Z8: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,F2,Z8)),aa(A,A,G,Z8))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_gfp(A,F2)),complete_lattice_gfp(A,G)) ) ) ).

% gfp_mono
tff(fact_7641_gfp__upperbound,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X4: A,F2: fun(A,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,F2,X4))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),complete_lattice_gfp(A,F2)) ) ) ).

% gfp_upperbound
tff(fact_7642_gfp__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),X4: A] :
          ( ! [U3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U3),aa(A,A,F2,U3))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U3),X4) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_gfp(A,F2)),X4) ) ) ).

% gfp_least
tff(fact_7643_gfp__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),Xa: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F3)
         => ( ( aa(A,A,F3,Xa) = Xa )
           => ( ! [Z2: A] :
                  ( ( aa(A,A,F3,Z2) = Z2 )
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z2),Xa) )
             => ( complete_lattice_gfp(A,F3) = Xa ) ) ) ) ) ).

% gfp_eqI
tff(fact_7644_gfp__def,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A)] : complete_lattice_gfp(A,F2) = aa(set(A),A,complete_Sup_Sup(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_ajo(fun(A,A),fun(A,$o),F2))) ) ).

% gfp_def
tff(fact_7645_def__Collect__coinduct,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),fun(A,$o)),A3: A,X4: set(A)] :
      ( ( A4 = complete_lattice_gfp(set(A),aTP_Lamp_ajp(fun(set(A),fun(A,$o)),fun(set(A),set(A)),P)) )
     => ( aa(fun(set(A),set(A)),$o,order_mono(set(A),set(A)),aTP_Lamp_ajp(fun(set(A),fun(A,$o)),fun(set(A),set(A)),P))
       => ( member(A,A3,X4)
         => ( ! [Z2: A] :
                ( member(A,Z2,X4)
               => aa(A,$o,aa(set(A),fun(A,$o),P,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X4),A4)),Z2) )
           => member(A,A3,A4) ) ) ) ) ).

% def_Collect_coinduct
tff(fact_7646_gfp__fun__UnI2,axiom,
    ! [A: $tType,F2: fun(set(A),set(A)),A3: A,X4: set(A)] :
      ( aa(fun(set(A),set(A)),$o,order_mono(set(A),set(A)),F2)
     => ( member(A,A3,complete_lattice_gfp(set(A),F2))
       => member(A,A3,aa(set(A),set(A),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X4),complete_lattice_gfp(set(A),F2)))) ) ) ).

% gfp_fun_UnI2
tff(fact_7647_weak__coinduct__image,axiom,
    ! [A: $tType,B: $tType,A3: A,X4: set(A),G: fun(A,B),F2: fun(set(B),set(B))] :
      ( member(A,A3,X4)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,G),X4)),aa(set(B),set(B),F2,aa(set(A),set(B),image2(A,B,G),X4)))
       => member(B,aa(A,B,G,A3),complete_lattice_gfp(set(B),F2)) ) ) ).

% weak_coinduct_image
tff(fact_7648_coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),X4: A] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),complete_lattice_gfp(A,F2))))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),complete_lattice_gfp(A,F2)) ) ) ) ).

% coinduct
tff(fact_7649_def__coinduct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A4: A,F2: fun(A,A),X4: A] :
          ( ( A4 = complete_lattice_gfp(A,F2) )
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),A4)))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),A4) ) ) ) ) ).

% def_coinduct
tff(fact_7650_coinduct__lemma,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X4: A,F2: fun(A,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X4),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),complete_lattice_gfp(A,F2))))
         => ( aa(fun(A,A),$o,order_mono(A,A),F2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),complete_lattice_gfp(A,F2))),aa(A,A,F2,aa(A,A,aa(A,fun(A,A),sup_sup(A),X4),complete_lattice_gfp(A,F2)))) ) ) ) ).

% coinduct_lemma
tff(fact_7651_coinduct__set,axiom,
    ! [A: $tType,F2: fun(set(A),set(A)),A3: A,X4: set(A)] :
      ( aa(fun(set(A),set(A)),$o,order_mono(set(A),set(A)),F2)
     => ( member(A,A3,X4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(A),set(A),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X4),complete_lattice_gfp(set(A),F2))))
         => member(A,A3,complete_lattice_gfp(set(A),F2)) ) ) ) ).

% coinduct_set
tff(fact_7652_def__coinduct__set,axiom,
    ! [A: $tType,A4: set(A),F2: fun(set(A),set(A)),A3: A,X4: set(A)] :
      ( ( A4 = complete_lattice_gfp(set(A),F2) )
     => ( aa(fun(set(A),set(A)),$o,order_mono(set(A),set(A)),F2)
       => ( member(A,A3,X4)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(A),set(A),F2,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),X4),A4)))
           => member(A,A3,A4) ) ) ) ) ).

% def_coinduct_set
tff(fact_7653_gfp__ordinal__induct,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),P: fun(A,$o)] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( ! [S2: A] :
                ( aa(A,$o,P,S2)
               => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_gfp(A,F2)),S2)
                 => aa(A,$o,P,aa(A,A,F2,S2)) ) )
           => ( ! [M9: set(A)] :
                  ( ! [X2: A] :
                      ( member(A,X2,M9)
                     => aa(A,$o,P,X2) )
                 => aa(A,$o,P,aa(set(A),A,complete_Inf_Inf(A),M9)) )
             => aa(A,$o,P,complete_lattice_gfp(A,F2)) ) ) ) ) ).

% gfp_ordinal_induct
tff(fact_7654_gfp__funpow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),Na: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( complete_lattice_gfp(A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,Na)),F2)) = complete_lattice_gfp(A,F2) ) ) ) ).

% gfp_funpow
tff(fact_7655_lfp__le__gfp,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A)] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),complete_lattice_lfp(A,F2)),complete_lattice_gfp(A,F2)) ) ) ).

% lfp_le_gfp
tff(fact_7656_gfp__Kleene__iter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F2: fun(A,A),K: nat] :
          ( aa(fun(A,A),$o,order_mono(A,A),F2)
         => ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,K)),F2),top_top(A)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A)) )
           => ( complete_lattice_gfp(A,F2) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F2),top_top(A)) ) ) ) ) ).

% gfp_Kleene_iter
tff(fact_7657_coinduct3,axiom,
    ! [A: $tType,F2: fun(set(A),set(A)),A3: A,X4: set(A)] :
      ( aa(fun(set(A),set(A)),$o,order_mono(set(A),set(A)),F2)
     => ( member(A,A3,X4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(A),set(A),F2,complete_lattice_lfp(set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_ajq(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),F2),X4))))
         => member(A,A3,complete_lattice_gfp(set(A),F2)) ) ) ) ).

% coinduct3
tff(fact_7658_gfp__transfer__bounded,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [P: fun(A,$o),F2: fun(A,A),Alpha: fun(A,B),G: fun(B,B)] :
          ( aa(A,$o,P,aa(A,A,F2,top_top(A)))
         => ( ! [X3: A] :
                ( aa(A,$o,P,X3)
               => aa(A,$o,P,aa(A,A,F2,X3)) )
           => ( ! [M9: fun(nat,A)] :
                  ( order_antimono(nat,A,M9)
                 => ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M9,I3))
                   => aa(A,$o,P,aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image2(nat,A,M9),top_top(set(nat))))) ) )
             => ( ! [M9: fun(nat,A)] :
                    ( order_antimono(nat,A,M9)
                   => ( ! [I3: nat] : aa(A,$o,P,aa(nat,A,M9,I3))
                     => ( aa(A,B,Alpha,aa(set(A),A,complete_Inf_Inf(A),aa(set(nat),set(A),image2(nat,A,M9),top_top(set(nat))))) = aa(set(B),B,complete_Inf_Inf(B),aa(set(nat),set(B),image2(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aet(fun(A,B),fun(fun(nat,A),fun(nat,B)),Alpha),M9)),top_top(set(nat)))) ) ) )
               => ( order_inf_continuous(A,A,F2)
                 => ( order_inf_continuous(B,B,G)
                   => ( ! [X3: A] :
                          ( aa(A,$o,P,X3)
                         => ( aa(A,B,Alpha,aa(A,A,F2,X3)) = aa(B,B,G,aa(A,B,Alpha,X3)) ) )
                     => ( ! [X3: B] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,G,X3)),aa(A,B,Alpha,aa(A,A,F2,top_top(A))))
                       => ( aa(A,B,Alpha,complete_lattice_gfp(A,F2)) = complete_lattice_gfp(B,G) ) ) ) ) ) ) ) ) ) ) ).

% gfp_transfer_bounded
tff(fact_7659_these__insert__Some,axiom,
    ! [A: $tType,Xa: A,A4: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),aa(A,option(A),some(A),Xa)),A4)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),these(A,A4)) ).

% these_insert_Some
tff(fact_7660_these__empty,axiom,
    ! [A: $tType] : these(A,bot_bot(set(option(A)))) = bot_bot(set(A)) ).

% these_empty
tff(fact_7661_inf__continuous__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & counta4013691401010221786attice(B) )
     => ! [P: fun(A,B),Q: fun(A,B)] :
          ( order_inf_continuous(A,B,P)
         => ( order_inf_continuous(A,B,Q)
           => order_inf_continuous(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajr(fun(A,B),fun(fun(A,B),fun(A,B)),P),Q)) ) ) ) ).

% inf_continuous_sup
tff(fact_7662_these__empty__eq,axiom,
    ! [A: $tType,B2: set(option(A))] :
      ( ( these(A,B2) = bot_bot(set(A)) )
    <=> ( ( B2 = bot_bot(set(option(A))) )
        | ( B2 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_7663_these__not__empty__eq,axiom,
    ! [A: $tType,B2: set(option(A))] :
      ( ( these(A,B2) != bot_bot(set(A)) )
    <=> ( ( B2 != bot_bot(set(option(A))) )
        & ( B2 != aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert2(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_7664_Max_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_max(A),aTP_Lamp_ta(A,fun(A,$o)),aTP_Lamp_ajs(A,fun(A,$o))) ) ).

% Max.semilattice_order_set_axioms
tff(fact_7665_Gcd__fin__def,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ( semiring_gcd_Gcd_fin(A) = bounde2362111253966948842tice_F(A,gcd_gcd(A),zero_zero(A),one_one(A)) ) ) ).

% Gcd_fin_def
tff(fact_7666_Inf__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => lattic4895041142388067077er_set(A,inf_inf(A),ord_less_eq(A),ord_less(A)) ) ).

% Inf_fin.semilattice_order_set_axioms
tff(fact_7667_Min_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic4895041142388067077er_set(A,ord_min(A),ord_less_eq(A),ord_less(A)) ) ).

% Min.semilattice_order_set_axioms
tff(fact_7668_Sup__fin_Osemilattice__order__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic4895041142388067077er_set(A,sup_sup(A),aTP_Lamp_ajt(A,fun(A,$o)),aTP_Lamp_aju(A,fun(A,$o))) ) ).

% Sup_fin.semilattice_order_set_axioms
tff(fact_7669_bounded__quasi__semilattice__set_Oremove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( member(A,A3,A4)
       => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A4) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ) ).

% bounded_quasi_semilattice_set.remove
tff(fact_7670_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),bot_bot(set(A)))))) ) ) ).

% bounded_quasi_semilattice_set.insert_remove
tff(fact_7671_bounded__quasi__semilattice__set_Oempty,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),bot_bot(set(A))) = Top ) ) ).

% bounded_quasi_semilattice_set.empty
tff(fact_7672_bounded__quasi__semilattice__set_Oinfinite,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A4) = Bot ) ) ) ).

% bounded_quasi_semilattice_set.infinite
tff(fact_7673_bounded__quasi__semilattice__set_Osubset,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),B2: set(A),A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
       => ( aa(A,A,aa(A,fun(A,A),F2,aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),B2)),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A4)) = aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A4) ) ) ) ).

% bounded_quasi_semilattice_set.subset
tff(fact_7674_bounded__quasi__semilattice__set_Oinsert,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A3: A,A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),A4)) = aa(A,A,aa(A,fun(A,A),F2,A3),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A4)) ) ) ).

% bounded_quasi_semilattice_set.insert
tff(fact_7675_bounded__quasi__semilattice__set_Ounion,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A4: set(A),B2: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),F2,aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A4)),aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),B2)) ) ) ).

% bounded_quasi_semilattice_set.union
tff(fact_7676_bounded__quasi__semilattice__set_Oeq__fold,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Top: A,Bot: A,Normalize: fun(A,A),A4: set(A)] :
      ( bounde6485984586167503788ce_set(A,F2,Top,Bot,Normalize)
     => ( aa(set(A),A,bounde2362111253966948842tice_F(A,F2,Top,Bot),A4) = $ite(aa(set(A),$o,finite_finite2(A),A4),finite_fold(A,A,F2,Top,A4),Bot) ) ) ).

% bounded_quasi_semilattice_set.eq_fold
tff(fact_7677_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Na: nat,M: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),Na)
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Na),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_7678_compute__powr__real,axiom,
    ! [B3: real,I: real] :
      powr_real(B3,I) = $ite(
        aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),zero_zero(real)),
        abort(real,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$false,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($true,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,zero_zero(literal)))))))))))))))))))))))))))))))),aa(real,fun(product_unit,real),aTP_Lamp_ajv(real,fun(real,fun(product_unit,real)),B3),I)),
        $ite(
          aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,I)) = I,
          $ite(aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),I),aa(nat,real,power_power(real,B3),aa(int,nat,nat2,archim6421214686448440834_floor(real,I))),divide_divide(real,one_one(real),aa(nat,real,power_power(real,B3),aa(int,nat,nat2,archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),I)))))),
          abort(real,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$false,$true,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$true,$false,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$false,$false,$false,$false,$true,$true,literal2($false,$false,$true,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$true,$true,$false,$true,$true,$true,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($false,$false,$false,$true,$false,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$true,$false,$true,$false,literal2($true,$false,$false,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($true,$true,$true,$false,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$false,$false,$true,$true,$true,literal2($false,$false,$false,$false,$false,$true,$false,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$false,$false,$true,$true,$true,$true,literal2($false,$false,$false,$false,$true,$true,$true,literal2($true,$true,$true,$true,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($true,$false,$true,$false,$false,$true,$true,literal2($false,$true,$true,$true,$false,$true,$true,literal2($false,$false,$true,$false,$true,$true,$true,zero_zero(literal)))))))))))))))))))))))))))))))))))),aa(real,fun(product_unit,real),aTP_Lamp_ajv(real,fun(real,fun(product_unit,real)),B3),I)) ) ) ).

% compute_powr_real
tff(fact_7679_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image2(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_7680_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),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_7681_range__nat__of__char,axiom,
    aa(set(char),set(nat),image2(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% range_nat_of_char
tff(fact_7682_partition__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( partition(A,P,Xs) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Yes),No4) )
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Yes)),aa(list(A),set(A),set2(A),No4)) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% partition_set
tff(fact_7683_nat__of__char__less__256,axiom,
    ! [C3: char] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C3)),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% nat_of_char_less_256
tff(fact_7684_length__code,axiom,
    ! [A: $tType] : size_size(list(A)) = gen_length(A,zero_zero(nat)) ).

% length_code
tff(fact_7685_card__def,axiom,
    ! [A: $tType] : finite_card(A) = finite_folding_F(A,nat,aTP_Lamp_mh(A,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_7686_folding__on_OF_Ocong,axiom,
    ! [A: $tType,B: $tType,F2: fun(A,fun(B,B)),Z: B] : finite_folding_F(A,B,F2,Z) = finite_folding_F(A,B,F2,Z) ).

% folding_on.F.cong
tff(fact_7687_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_7688_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),Xa: A,Z: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( member(A,Xa,A4)
           => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A4) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_7689_folding__on__def,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S,F2)
    <=> ! [X: A,Y4: A] :
          ( member(A,X,S)
         => ( member(A,Y4,S)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),aa(A,fun(B,B),F2,X)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,Y4)) ) ) ) ) ).

% folding_on_def
tff(fact_7690_folding__on_Ocomp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Ya: A] :
      ( finite_folding_on(A,B,S,F2)
     => ( member(A,Xa,S)
       => ( member(A,Ya,S)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Ya)),aa(A,fun(B,B),F2,Xa)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xa)),aa(A,fun(B,B),F2,Ya)) ) ) ) ) ).

% folding_on.comp_fun_commute_on
tff(fact_7691_folding__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,S)
         => ( member(A,Y,S)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y)) ) ) )
     => finite_folding_on(A,B,S,F2) ) ).

% folding_on.intro
tff(fact_7692_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_mh(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_7693_folding__on_Oempty,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),Z: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),bot_bot(set(A))) = Z ) ) ).

% folding_on.empty
tff(fact_7694_folding__on_Oinfinite,axiom,
    ! [A: $tType,B: $tType,S: set(A),F2: fun(A,fun(B,B)),A4: set(A),Z: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A4) = Z ) ) ) ).

% folding_on.infinite
tff(fact_7695_folding__on_Oeq__fold,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Z: B,A4: set(A)] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),A4) = finite_fold(A,B,F2,Z,A4) ) ) ).

% folding_on.eq_fold
tff(fact_7696_folding__on_Oinsert,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite_folding_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( ~ member(A,Xa,A4)
           => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(set(A),B,finite_folding_F(A,B,F2,Z),A4)) ) ) ) ) ) ).

% folding_on.insert
tff(fact_7697_folding__idem__on_Oinsert__idem,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,A4: set(A),Z: B] :
      ( finite1890593828518410140dem_on(A,B,S,F2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)),S)
       => ( aa(set(A),$o,finite_finite2(A),A4)
         => ( aa(set(A),B,finite_folding_F(A,B,F2,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(B,B,aa(A,fun(B,B),F2,Xa),aa(set(A),B,finite_folding_F(A,B,F2,Z),A4)) ) ) ) ) ).

% folding_idem_on.insert_idem
tff(fact_7698_rel__fun__iff__geq__image2p,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,R: fun(A,fun(B,$o)),S: fun(C,fun(D,$o)),F2: fun(A,C),G: fun(B,D)] :
      ( aa(fun(B,D),$o,aa(fun(A,C),fun(fun(B,D),$o),bNF_rel_fun(A,B,C,D,R,S),F2),G)
    <=> aa(fun(C,fun(D,$o)),$o,aa(fun(C,fun(D,$o)),fun(fun(C,fun(D,$o)),$o),ord_less_eq(fun(C,fun(D,$o))),bNF_Greatest_image2p(A,C,B,D,F2,G,R)),S) ) ).

% rel_fun_iff_geq_image2p
tff(fact_7699_folding__idem__on_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S,F2)
     => finite_folding_on(A,B,S,F2) ) ).

% folding_idem_on.axioms(1)
tff(fact_7700_folding__idem__on_Ocomp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B)),Xa: A,Ya: A] :
      ( finite1890593828518410140dem_on(A,B,S,F2)
     => ( member(A,Xa,S)
       => ( member(A,Ya,S)
         => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xa)),aa(A,fun(B,B),F2,Xa)) = aa(A,fun(B,B),F2,Xa) ) ) ) ) ).

% folding_idem_on.comp_fun_idem_on
tff(fact_7701_folding__idem__on_Ointro,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite_folding_on(A,B,S,F2)
     => ( finite6916993218817215295axioms(A,B,S,F2)
       => finite1890593828518410140dem_on(A,B,S,F2) ) ) ).

% folding_idem_on.intro
tff(fact_7702_folding__idem__on__def,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S,F2)
    <=> ( finite_folding_on(A,B,S,F2)
        & finite6916993218817215295axioms(A,B,S,F2) ) ) ).

% folding_idem_on_def
tff(fact_7703_folding__idem__on__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( ! [X3: A,Y: A] :
          ( member(A,X3,S)
         => ( member(A,Y,S)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,X3)) = aa(A,fun(B,B),F2,X3) ) ) )
     => finite6916993218817215295axioms(A,B,S,F2) ) ).

% folding_idem_on_axioms.intro
tff(fact_7704_folding__idem__on__axioms__def,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite6916993218817215295axioms(A,B,S,F2)
    <=> ! [X: A,Y4: A] :
          ( member(A,X,S)
         => ( member(A,Y4,S)
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,X)) = aa(A,fun(B,B),F2,X) ) ) ) ) ).

% folding_idem_on_axioms_def
tff(fact_7705_folding__idem__on_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,S: set(A),F2: fun(A,fun(B,B))] :
      ( finite1890593828518410140dem_on(A,B,S,F2)
     => finite6916993218817215295axioms(A,B,S,F2) ) ).

% folding_idem_on.axioms(2)
tff(fact_7706_folding__idem__def_H,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite_folding_idem(A,B,F2)
    <=> finite1890593828518410140dem_on(A,B,top_top(set(A)),F2) ) ).

% folding_idem_def'
tff(fact_7707_Func__empty,axiom,
    ! [B: $tType,A: $tType,B2: set(B)] : bNF_Wellorder_Func(A,B,bot_bot(set(A)),B2) = aa(set(fun(A,B)),set(fun(A,B)),aa(fun(A,B),fun(set(fun(A,B)),set(fun(A,B))),insert2(fun(A,B)),aTP_Lamp_ajw(A,B)),bot_bot(set(fun(A,B)))) ).

% Func_empty
tff(fact_7708_folding__idem_Ocomp__fun__idem,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),Xa: A] :
      ( finite_folding_idem(A,B,F2)
     => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xa)),aa(A,fun(B,B),F2,Xa)) = aa(A,fun(B,B),F2,Xa) ) ) ).

% folding_idem.comp_fun_idem
tff(fact_7709_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xa: fun(A,B),Xaa: list(A),Ya: A] :
          ( ( arg_min_list(A,B,Xa,Xaa) = Ya )
         => ( ! [X3: A] :
                ( ( Xaa = aa(list(A),list(A),cons(A,X3),nil(A)) )
               => ( Ya != X3 ) )
           => ( ! [X3: A,Y: A,Zs2: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y),Zs2)) )
                 => ( Ya != $let(
                        m: A,
                        m:= arg_min_list(A,B,Xa,aa(list(A),list(A),cons(A,Y),Zs2)),
                        $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Xa,X3)),aa(A,B,Xa,m)),X3,m) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ( Ya != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_7710_prod__filtermap1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F2: fun(C,A),F3: filter(C),G2: filter(B)] : prod_filter(A,B,aa(filter(C),filter(A),aa(fun(C,A),fun(filter(C),filter(A)),filtermap(C,A),F2),F3),G2) = aa(filter(product_prod(C,B)),filter(product_prod(A,B)),aa(fun(product_prod(C,B),product_prod(A,B)),fun(filter(product_prod(C,B)),filter(product_prod(A,B))),filtermap(product_prod(C,B),product_prod(A,B)),product_apfst(C,A,B,F2)),prod_filter(C,B,F3,G2)) ).

% prod_filtermap1
tff(fact_7711_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F2: fun(A,B),Xa: A,Ya: A,Zs: list(A)] :
          arg_min_list(A,B,F2,aa(list(A),list(A),cons(A,Xa),aa(list(A),list(A),cons(A,Ya),Zs))) = $let(
            m: A,
            m:= arg_min_list(A,B,F2,aa(list(A),list(A),cons(A,Ya),Zs)),
            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F2,Xa)),aa(A,B,F2,m)),Xa,m) ) ) ).

% arg_min_list.simps(2)
tff(fact_7712_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xa: fun(A,B),Xaa: list(A),Ya: A] :
          ( ( arg_min_list(A,B,Xa,Xaa) = Ya )
         => ( aa(product_prod(fun(A,B),list(A)),$o,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),Xa),Xaa))
           => ( ! [X3: A] :
                  ( ( Xaa = aa(list(A),list(A),cons(A,X3),nil(A)) )
                 => ( ( Ya = X3 )
                   => ~ aa(product_prod(fun(A,B),list(A)),$o,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),Xa),aa(list(A),list(A),cons(A,X3),nil(A)))) ) )
             => ( ! [X3: A,Y: A,Zs2: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y),Zs2)) )
                   => ( ( Ya = $let(
                            m: A,
                            m:= arg_min_list(A,B,Xa,aa(list(A),list(A),cons(A,Y),Zs2)),
                            $ite(aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Xa,X3)),aa(A,B,Xa,m)),X3,m) ) )
                     => ~ aa(product_prod(fun(A,B),list(A)),$o,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),Xa),aa(list(A),list(A),cons(A,X3),aa(list(A),list(A),cons(A,Y),Zs2)))) ) )
               => ~ ( ( Xaa = nil(A) )
                   => ( ( Ya = undefined(A) )
                     => ~ aa(product_prod(fun(A,B),list(A)),$o,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),Xa),nil(A))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_7713_single__valuedp__single__valued__eq,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( single_valuedp(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),R2))
    <=> single_valued(A,B,R2) ) ).

% single_valuedp_single_valued_eq
tff(fact_7714_single__valuedp__less__eq,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o)),S3: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),R2),S3)
     => ( single_valuedp(A,B,S3)
       => single_valuedp(A,B,R2) ) ) ).

% single_valuedp_less_eq
tff(fact_7715_single__valuedp__def,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o))] :
      ( single_valuedp(A,B,R2)
    <=> ! [X: A,Y4: B] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,X),Y4)
         => ! [Z5: B] :
              ( aa(B,$o,aa(A,fun(B,$o),R2,X),Z5)
             => ( Y4 = Z5 ) ) ) ) ).

% single_valuedp_def
tff(fact_7716_single__valuedpI,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
      ( ! [X3: A,Y: B,Z2: B] :
          ( aa(B,$o,aa(A,fun(B,$o),R2,X3),Y)
         => ( aa(B,$o,aa(A,fun(B,$o),R2,X3),Z2)
           => ( Y = Z2 ) ) )
     => single_valuedp(A,B,R2) ) ).

% single_valuedpI
tff(fact_7717_single__valuedpD,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),Xa: A,Ya: B,Z: B] :
      ( single_valuedp(A,B,R2)
     => ( aa(B,$o,aa(A,fun(B,$o),R2,Xa),Ya)
       => ( aa(B,$o,aa(A,fun(B,$o),R2,Xa),Z)
         => ( Ya = Z ) ) ) ) ).

% single_valuedpD
tff(fact_7718_single__valuedp__bot,axiom,
    ! [B: $tType,A: $tType] : single_valuedp(A,B,bot_bot(fun(A,fun(B,$o)))) ).

% single_valuedp_bot
tff(fact_7719_eq__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),aTP_Lamp_ajx(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P)) ).

% eq_subset
tff(fact_7720_less__than__iff,axiom,
    ! [Xa: nat,Ya: nat] :
      ( member(product_prod(nat,nat),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),Ya),less_than)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Xa),Ya) ) ).

% less_than_iff
tff(fact_7721_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),R: $o,Xa: A,Ya: B] :
      ( ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
        & (R) )
     => ( (R)
        & ( aa(B,$o,aa(A,fun(B,$o),P,Xa),Ya)
         => aa(B,$o,aa(A,fun(B,$o),Q,Xa),Ya) ) ) ) ).

% predicate2D_conj
tff(fact_7722_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F2: fun(fun(A,B),C),G: C] :
          ( ! [X3: fun(A,B)] : aa(fun(A,B),C,F2,X3) = G
         => ( aa(fun(A,B),C,F2,aTP_Lamp_ajy(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_7723_finite__subset__Union__chain,axiom,
    ! [A: $tType,A4: set(A),B12: set(set(A)),A18: set(set(A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12))
       => ( ( B12 != bot_bot(set(set(A))) )
         => ( aa(set(set(A)),$o,pred_chain(set(A),A18,ord_less(set(A))),B12)
           => ~ ! [B4: set(A)] :
                  ( member(set(A),B4,B12)
                 => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B4) ) ) ) ) ) ).

% finite_subset_Union_chain
tff(fact_7724_elimnum,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Na)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extended_enat2(aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Na))) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimnum
tff(fact_7725_enat__ord__simps_I2_J,axiom,
    ! [M: nat,Na: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(M)),extended_enat2(Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M),Na) ) ).

% enat_ord_simps(2)
tff(fact_7726_enat__ord__simps_I1_J,axiom,
    ! [M: nat,Na: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_enat2(M)),extended_enat2(Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Na) ) ).

% enat_ord_simps(1)
tff(fact_7727_idiff__enat__0__right,axiom,
    ! [Na: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,Na),extended_enat2(zero_zero(nat))) = Na ).

% idiff_enat_0_right
tff(fact_7728_idiff__enat__0,axiom,
    ! [Na: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,extended_enat2(zero_zero(nat))),Na) = extended_enat2(zero_zero(nat)) ).

% idiff_enat_0
tff(fact_7729_numeral__less__enat__iff,axiom,
    ! [M: num,Na: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),extended_enat2(Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),Na) ) ).

% numeral_less_enat_iff
tff(fact_7730_numeral__le__enat__iff,axiom,
    ! [M: num,Na: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),extended_enat2(Na))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),Na) ) ).

% numeral_le_enat_iff
tff(fact_7731_subset__Zorn_H,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ! [C7: set(set(A))] :
          ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C7)
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C7),A4) )
     => ? [X3: set(A)] :
          ( member(set(A),X3,A4)
          & ! [Xa2: set(A)] :
              ( member(set(A),Xa2,A4)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa2)
               => ( Xa2 = X3 ) ) ) ) ) ).

% subset_Zorn'
tff(fact_7732_pred__on_OchainI,axiom,
    ! [A: $tType,C2: set(A),A4: set(A),P: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),A4)
     => ( ! [X3: A,Y: A] :
            ( member(A,X3,C2)
           => ( member(A,Y,C2)
             => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X3),Y)
                | aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),Y),X3) ) ) )
       => aa(set(A),$o,pred_chain(A,A4,P),C2) ) ) ).

% pred_on.chainI
tff(fact_7733_pred__on_Ochain__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C2: set(A)] :
      ( aa(set(A),$o,pred_chain(A,A4,P),C2)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C2),A4)
        & ! [X: A] :
            ( member(A,X,C2)
           => ! [Xa3: A] :
                ( member(A,Xa3,C2)
               => ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X),Xa3)
                  | aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),Xa3),X) ) ) ) ) ) ).

% pred_on.chain_def
tff(fact_7734_subset_Ochain__def,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C2),A4)
        & ! [X: set(A)] :
            ( member(set(A),X,C2)
           => ! [Xa3: set(A)] :
                ( member(set(A),Xa3,C2)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X),Xa3)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Xa3),X) ) ) ) ) ) ).

% subset.chain_def
tff(fact_7735_subset_OchainI,axiom,
    ! [A: $tType,C2: set(set(A)),A4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C2),A4)
     => ( ! [X3: set(A),Y: set(A)] :
            ( member(set(A),X3,C2)
           => ( member(set(A),Y,C2)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X3),Y)
                | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Y),X3) ) ) )
       => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2) ) ) ).

% subset.chainI
tff(fact_7736_finite__enat__bounded,axiom,
    ! [A4: set(extended_enat),Na: nat] :
      ( ! [Y: extended_enat] :
          ( member(extended_enat,Y,A4)
         => aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Y),extended_enat2(Na)) )
     => aa(set(extended_enat),$o,finite_finite2(extended_enat),A4) ) ).

% finite_enat_bounded
tff(fact_7737_enat__ile,axiom,
    ! [Na: extended_enat,M: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Na),extended_enat2(M))
     => ? [K2: nat] : Na = extended_enat2(K2) ) ).

% enat_ile
tff(fact_7738_subset__Zorn,axiom,
    ! [A: $tType,A4: set(set(A))] :
      ( ! [C7: set(set(A))] :
          ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C7)
         => ? [X2: set(A)] :
              ( member(set(A),X2,A4)
              & ! [Xa4: set(A)] :
                  ( member(set(A),Xa4,C7)
                 => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa4),X2) ) ) )
     => ? [X3: set(A)] :
          ( member(set(A),X3,A4)
          & ! [Xa2: set(A)] :
              ( member(set(A),Xa2,A4)
             => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa2)
               => ( Xa2 = X3 ) ) ) ) ) ).

% subset_Zorn
tff(fact_7739_chains__alt__def,axiom,
    ! [A: $tType,A4: set(set(A))] : chains2(A,A4) = aa(fun(set(set(A)),$o),set(set(set(A))),collect(set(set(A))),pred_chain(set(A),A4,ord_less(set(A)))) ).

% chains_alt_def
tff(fact_7740_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Uu: extended_enat] : vEBT_VEBT_elim_dead(vEBT_Leaf((A3),(B3)),Uu) = vEBT_Leaf((A3),(B3)) ).

% VEBT_internal.elim_dead.simps(1)
tff(fact_7741_chain__incr,axiom,
    ! [A: $tType,Y3: fun(A,extended_enat),K: nat] :
      ( ! [I2: A] :
        ? [J5: A] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(A,extended_enat,Y3,I2)),aa(A,extended_enat,Y3,J5))
     => ? [J2: A] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(K)),aa(A,extended_enat,Y3,J2)) ) ).

% chain_incr
tff(fact_7742_enat__iless,axiom,
    ! [Na: extended_enat,M: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),extended_enat2(M))
     => ? [K2: nat] : Na = extended_enat2(K2) ) ).

% enat_iless
tff(fact_7743_less__enatE,axiom,
    ! [Na: extended_enat,M: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),extended_enat2(M))
     => ~ ! [K2: nat] :
            ( ( Na = extended_enat2(K2) )
           => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),K2),M) ) ) ).

% less_enatE
tff(fact_7744_subset_Ochain__total,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A)),Xa: set(A),Ya: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2)
     => ( member(set(A),Xa,C2)
       => ( member(set(A),Ya,C2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Xa),Ya)
            | aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),Ya),Xa) ) ) ) ) ).

% subset.chain_total
tff(fact_7745_subset__chain__def,axiom,
    ! [A: $tType,A18: set(set(A)),C10: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A18,ord_less(set(A))),C10)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),C10),A18)
        & ! [X: set(A)] :
            ( member(set(A),X,C10)
           => ! [Xa3: set(A)] :
                ( member(set(A),Xa3,C10)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),Xa3)
                  | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Xa3),X) ) ) ) ) ) ).

% subset_chain_def
tff(fact_7746_Suc__ile__eq,axiom,
    ! [M: nat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_enat2(aa(nat,nat,suc,M))),Na)
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(M)),Na) ) ).

% Suc_ile_eq
tff(fact_7747_pred__on_Ochain__empty,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o))] : aa(set(A),$o,pred_chain(A,A4,P),bot_bot(set(A))) ).

% pred_on.chain_empty
tff(fact_7748_subset_Ochain__empty,axiom,
    ! [A: $tType,A4: set(set(A))] : aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),bot_bot(set(set(A)))) ).

% subset.chain_empty
tff(fact_7749_zero__enat__def,axiom,
    zero_zero(extended_enat) = extended_enat2(zero_zero(nat)) ).

% zero_enat_def
tff(fact_7750_enat__0__iff_I1_J,axiom,
    ! [Xa: nat] :
      ( ( extended_enat2(Xa) = zero_zero(extended_enat) )
    <=> ( Xa = zero_zero(nat) ) ) ).

% enat_0_iff(1)
tff(fact_7751_enat__0__iff_I2_J,axiom,
    ! [Xa: nat] :
      ( ( zero_zero(extended_enat) = extended_enat2(Xa) )
    <=> ( Xa = zero_zero(nat) ) ) ).

% enat_0_iff(2)
tff(fact_7752_subset__chain__insert,axiom,
    ! [A: $tType,A18: set(set(A)),B2: set(A),B12: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A18,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),B2),B12))
    <=> ( member(set(A),B2,A18)
        & ! [X: set(A)] :
            ( member(set(A),X,B12)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X),B2)
              | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),X) ) )
        & aa(set(set(A)),$o,pred_chain(set(A),A18,ord_less(set(A))),B12) ) ) ).

% subset_chain_insert
tff(fact_7753_chain__subset__alt__def,axiom,
    ! [A: $tType,C2: set(set(A))] :
      ( chain_subset(A,C2)
    <=> aa(set(set(A)),$o,pred_chain(set(A),top_top(set(set(A))),ord_less(set(A))),C2) ) ).

% chain_subset_alt_def
tff(fact_7754_iadd__le__enat__iff,axiom,
    ! [Xa: extended_enat,Ya: extended_enat,Na: nat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xa),Ya)),extended_enat2(Na))
    <=> ? [Y7: nat,X9: nat] :
          ( ( Xa = extended_enat2(X9) )
          & ( Ya = extended_enat2(Y7) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X9),Y7)),Na) ) ) ).

% iadd_le_enat_iff
tff(fact_7755_subset__Zorn__nonempty,axiom,
    ! [A: $tType,A18: set(set(A))] :
      ( ( A18 != bot_bot(set(set(A))) )
     => ( ! [C11: set(set(A))] :
            ( ( C11 != bot_bot(set(set(A))) )
           => ( aa(set(set(A)),$o,pred_chain(set(A),A18,ord_less(set(A))),C11)
             => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C11),A18) ) )
       => ? [X3: set(A)] :
            ( member(set(A),X3,A18)
            & ! [Xa2: set(A)] :
                ( member(set(A),Xa2,A18)
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X3),Xa2)
                 => ( Xa2 = X3 ) ) ) ) ) ) ).

% subset_Zorn_nonempty
tff(fact_7756_Union__in__chain,axiom,
    ! [A: $tType,B12: set(set(A)),A18: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),B12)
     => ( ( B12 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A18,ord_less(set(A))),B12)
         => member(set(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B12),B12) ) ) ) ).

% Union_in_chain
tff(fact_7757_Inter__in__chain,axiom,
    ! [A: $tType,B12: set(set(A)),A18: set(set(A))] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),B12)
     => ( ( B12 != bot_bot(set(set(A))) )
       => ( aa(set(set(A)),$o,pred_chain(set(A),A18,ord_less(set(A))),B12)
         => member(set(A),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B12),B12) ) ) ) ).

% Inter_in_chain
tff(fact_7758_subset_Ochain__extend,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A)),Z: set(A)] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2)
     => ( member(set(A),Z,A4)
       => ( ! [X3: set(A)] :
              ( member(set(A),X3,C2)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o)),aa(fun(set(A),fun(set(A),$o)),fun(fun(set(A),fun(set(A),$o)),fun(set(A),fun(set(A),$o))),sup_sup(fun(set(A),fun(set(A),$o))),ord_less(set(A))),fequal(set(A))),X3),Z) )
         => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),Z),bot_bot(set(set(A))))),C2)) ) ) ) ).

% subset.chain_extend
tff(fact_7759_pred__on_Ochain__extend,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C2: set(A),Z: A] :
      ( aa(set(A),$o,pred_chain(A,A4,P),C2)
     => ( member(A,Z,A4)
       => ( ! [X3: A] :
              ( member(A,X3,C2)
             => aa(A,$o,aa(A,fun(A,$o),aa(fun(A,fun(A,$o)),fun(A,fun(A,$o)),aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),fun(A,fun(A,$o))),sup_sup(fun(A,fun(A,$o))),P),fequal(A)),X3),Z) )
         => aa(set(A),$o,pred_chain(A,A4,P),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Z),bot_bot(set(A)))),C2)) ) ) ) ).

% pred_on.chain_extend
tff(fact_7760_Chains__subset_H,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( refl_on(A,top_top(set(A)),R2)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_ahx(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))),chains(A,R2)) ) ).

% Chains_subset'
tff(fact_7761_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: extended_enat,Ya: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xa,Xaa) = Ya )
     => ( ! [A5: $o,B5: $o] :
            ( ( Xa = vEBT_Leaf((A5),(B5)) )
           => ( Ya != vEBT_Leaf((A5),(B5)) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary3) )
             => ( ( Xaa = extend4730790105801354508finity(extended_enat) )
               => ( Ya != vEBT_Node(Info2,Deg2,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_ajz(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList2),vEBT_VEBT_elim_dead(Summary3,extend4730790105801354508finity(extended_enat))) ) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary3) )
               => ! [L6: nat] :
                    ( ( Xaa = extended_enat2(L6) )
                   => ( Ya != vEBT_Node(Info2,Deg2,take(vEBT_VEBT,divide_divide(nat,L6,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_ajz(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList2)),vEBT_VEBT_elim_dead(Summary3,extended_enat2(divide_divide(nat,L6,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
tff(fact_7762_elimcomplete,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Na: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeList,Summary),Na)
     => ( vEBT_VEBT_elim_dead(vEBT_Node(Info,Deg,TreeList,Summary),extend4730790105801354508finity(extended_enat)) = vEBT_Node(Info,Deg,TreeList,Summary) ) ) ).

% elimcomplete
tff(fact_7763_enat__ord__simps_I6_J,axiom,
    ! [Q5: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extend4730790105801354508finity(extended_enat)),Q5) ).

% enat_ord_simps(6)
tff(fact_7764_enat__ord__simps_I4_J,axiom,
    ! [Q5: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Q5),extend4730790105801354508finity(extended_enat))
    <=> ( Q5 != extend4730790105801354508finity(extended_enat) ) ) ).

% enat_ord_simps(4)
tff(fact_7765_enat__ord__code_I3_J,axiom,
    ! [Q5: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Q5),extend4730790105801354508finity(extended_enat)) ).

% enat_ord_code(3)
tff(fact_7766_enat__ord__simps_I5_J,axiom,
    ! [Q5: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extend4730790105801354508finity(extended_enat)),Q5)
    <=> ( Q5 = extend4730790105801354508finity(extended_enat) ) ) ).

% enat_ord_simps(5)
tff(fact_7767_times__enat__simps_I4_J,axiom,
    ! [M: nat] :
      aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extended_enat2(M)),extend4730790105801354508finity(extended_enat)) = $ite(M = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)) ).

% times_enat_simps(4)
tff(fact_7768_times__enat__simps_I3_J,axiom,
    ! [Na: nat] :
      aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(Na)) = $ite(Na = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)) ).

% times_enat_simps(3)
tff(fact_7769_enat__ord__code_I5_J,axiom,
    ! [Na: nat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(Na)) ).

% enat_ord_code(5)
tff(fact_7770_infinity__ileE,axiom,
    ! [M: nat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(M)) ).

% infinity_ileE
tff(fact_7771_enat__ord__code_I4_J,axiom,
    ! [M: nat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(M)),extend4730790105801354508finity(extended_enat)) ).

% enat_ord_code(4)
tff(fact_7772_less__infinityE,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),extend4730790105801354508finity(extended_enat))
     => ~ ! [K2: nat] : Na != extended_enat2(K2) ) ).

% less_infinityE
tff(fact_7773_infinity__ilessE,axiom,
    ! [M: nat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extend4730790105801354508finity(extended_enat)),extended_enat2(M)) ).

% infinity_ilessE
tff(fact_7774_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [Xa: product_prod(vEBT_VEBT,extended_enat)] :
      ( ! [A5: $o,B5: $o,Uu2: extended_enat] : Xa != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Leaf((A5),(B5))),Uu2)
     => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] : Xa != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info2,Deg2,TreeList2,Summary3)),extend4730790105801354508finity(extended_enat))
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT,L6: nat] : Xa != aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info2,Deg2,TreeList2,Summary3)),extended_enat2(L6)) ) ) ).

% VEBT_internal.elim_dead.cases
tff(fact_7775_enat__add__left__cancel__less,axiom,
    ! [A3: extended_enat,B3: extended_enat,C3: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),B3)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),C3))
    <=> ( ( A3 != extend4730790105801354508finity(extended_enat) )
        & aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),B3),C3) ) ) ).

% enat_add_left_cancel_less
tff(fact_7776_enat__ord__simps_I3_J,axiom,
    ! [Q5: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Q5),extend4730790105801354508finity(extended_enat)) ).

% enat_ord_simps(3)
tff(fact_7777_enat__add__left__cancel__le,axiom,
    ! [A3: extended_enat,B3: extended_enat,C3: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),B3)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),A3),C3))
    <=> ( ( A3 = extend4730790105801354508finity(extended_enat) )
        | aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),B3),C3) ) ) ).

% enat_add_left_cancel_le
tff(fact_7778_imult__infinity__right,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),Na),extend4730790105801354508finity(extended_enat)) = extend4730790105801354508finity(extended_enat) ) ) ).

% imult_infinity_right
tff(fact_7779_imult__infinity,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),Na)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),extend4730790105801354508finity(extended_enat)),Na) = extend4730790105801354508finity(extended_enat) ) ) ).

% imult_infinity
tff(fact_7780_mono__Chains,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),R2),S3)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),chains(A,R2)),chains(A,S3)) ) ).

% mono_Chains
tff(fact_7781_Chains__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),chains(A,R2)),aa(fun(set(A),$o),set(set(A)),collect(set(A)),pred_chain(A,top_top(set(A)),aTP_Lamp_ahx(set(product_prod(A,A)),fun(A,fun(A,$o)),R2)))) ).

% Chains_subset
tff(fact_7782_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [Xa: vEBT_VEBT,Xaa: extended_enat,Ya: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead(Xa,Xaa) = Ya )
     => ( aa(product_prod(vEBT_VEBT,extended_enat),$o,accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel),aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,Xa),Xaa))
       => ( ! [A5: $o,B5: $o] :
              ( ( Xa = vEBT_Leaf((A5),(B5)) )
             => ( ( Ya = vEBT_Leaf((A5),(B5)) )
               => ~ aa(product_prod(vEBT_VEBT,extended_enat),$o,accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel),aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Leaf((A5),(B5))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary3) )
               => ( ( Xaa = extend4730790105801354508finity(extended_enat) )
                 => ( ( Ya = vEBT_Node(Info2,Deg2,map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_ajz(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList2),vEBT_VEBT_elim_dead(Summary3,extend4730790105801354508finity(extended_enat))) )
                   => ~ aa(product_prod(vEBT_VEBT,extended_enat),$o,accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel),aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info2,Deg2,TreeList2,Summary3)),extend4730790105801354508finity(extended_enat))) ) ) )
           => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList2: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                  ( ( Xa = vEBT_Node(Info2,Deg2,TreeList2,Summary3) )
                 => ! [L6: nat] :
                      ( ( Xaa = extended_enat2(L6) )
                     => ( ( Ya = vEBT_Node(Info2,Deg2,take(vEBT_VEBT,divide_divide(nat,L6,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2))))),map(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_ajz(nat,fun(vEBT_VEBT,vEBT_VEBT),Deg2),TreeList2)),vEBT_VEBT_elim_dead(Summary3,extended_enat2(divide_divide(nat,L6,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) )
                       => ~ aa(product_prod(vEBT_VEBT,extended_enat),$o,accp(product_prod(vEBT_VEBT,extended_enat),vEBT_V312737461966249ad_rel),aa(extended_enat,product_prod(vEBT_VEBT,extended_enat),product_Pair(vEBT_VEBT,extended_enat,vEBT_Node(Info2,Deg2,TreeList2,Summary3)),extended_enat2(L6))) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
tff(fact_7783_times__enat__def,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Na) = extended_case_enat(extended_enat,aTP_Lamp_akb(extended_enat,fun(nat,extended_enat),Na),
        $ite(Na = zero_zero(extended_enat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)),
        M) ).

% times_enat_def
tff(fact_7784_pred__on_Onot__maxchain__Some,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C2: set(A)] :
      ( aa(set(A),$o,pred_chain(A,A4,P),C2)
     => ( ~ pred_maxchain(A,A4,P,C2)
       => ( aa(set(A),$o,pred_chain(A,A4,P),fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_akc(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C2)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),C2),fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_akc(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C2))) ) ) ) ).

% pred_on.not_maxchain_Some
tff(fact_7785_Range__Union,axiom,
    ! [A: $tType,B: $tType,S: set(set(product_prod(B,A)))] : aa(set(product_prod(B,A)),set(A),range(B,A),aa(set(set(product_prod(B,A))),set(product_prod(B,A)),complete_Sup_Sup(set(product_prod(B,A))),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(B,A))),set(set(A)),image2(set(product_prod(B,A)),set(A),range(B,A)),S)) ).

% Range_Union
tff(fact_7786_Range__Id__on,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(A,A)),set(A),range(A,A),id_on(A,A4)) = A4 ).

% Range_Id_on
tff(fact_7787_Range__empty,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(A),range(B,A),bot_bot(set(product_prod(B,A)))) = bot_bot(set(A)) ).

% Range_empty
tff(fact_7788_Range__Id,axiom,
    ! [A: $tType] : aa(set(product_prod(A,A)),set(A),range(A,A),id2(A)) = top_top(set(A)) ).

% Range_Id
tff(fact_7789_Range__Collect__case__prod,axiom,
    ! [A: $tType,B: $tType,P: fun(B,fun(A,$o))] : aa(set(product_prod(B,A)),set(A),range(B,A),aa(fun(product_prod(B,A),$o),set(product_prod(B,A)),collect(product_prod(B,A)),aa(fun(B,fun(A,$o)),fun(product_prod(B,A),$o),product_case_prod(B,A,$o),P))) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_akd(fun(B,fun(A,$o)),fun(A,$o),P)) ).

% Range_Collect_case_prod
tff(fact_7790_Range__insert,axiom,
    ! [A: $tType,B: $tType,A3: B,B3: A,R2: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),product_Pair(B,A,A3),B3)),R2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),B3),aa(set(product_prod(B,A)),set(A),range(B,A),R2)) ).

% Range_insert
tff(fact_7791_Range_Ocases,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( member(A,A3,aa(set(product_prod(B,A)),set(A),range(B,A),R2))
     => ~ ! [A5: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,A5),A3),R2) ) ).

% Range.cases
tff(fact_7792_Range_Osimps,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( member(A,A3,aa(set(product_prod(B,A)),set(A),range(B,A),R2))
    <=> ? [A9: B,B7: A] :
          ( ( A3 = B7 )
          & member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,A9),B7),R2) ) ) ).

% Range.simps
tff(fact_7793_Range_Ointros,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),R2)
     => member(B,B3,aa(set(product_prod(A,B)),set(B),range(A,B),R2)) ) ).

% Range.intros
tff(fact_7794_RangeE,axiom,
    ! [A: $tType,B: $tType,B3: A,R2: set(product_prod(B,A))] :
      ( member(A,B3,aa(set(product_prod(B,A)),set(A),range(B,A),R2))
     => ~ ! [A5: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,A5),B3),R2) ) ).

% RangeE
tff(fact_7795_Range__iff,axiom,
    ! [A: $tType,B: $tType,A3: A,R2: set(product_prod(B,A))] :
      ( member(A,A3,aa(set(product_prod(B,A)),set(A),range(B,A),R2))
    <=> ? [Y4: B] : member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,Y4),A3),R2) ) ).

% Range_iff
tff(fact_7796_Range__empty__iff,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] :
      ( ( aa(set(product_prod(B,A)),set(A),range(B,A),R2) = bot_bot(set(A)) )
    <=> ( R2 = bot_bot(set(product_prod(B,A))) ) ) ).

% Range_empty_iff
tff(fact_7797_subset_OHausdorff,axiom,
    ! [A: $tType,A4: set(set(A))] :
    ? [X_1: set(set(A))] : pred_maxchain(set(A),A4,ord_less(set(A)),X_1) ).

% subset.Hausdorff
tff(fact_7798_Range__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S3)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(product_prod(A,B)),set(B),range(A,B),R2)),aa(set(product_prod(A,B)),set(B),range(A,B),S3)) ) ).

% Range_mono
tff(fact_7799_Range__snd,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range(B,A),R2) = aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),R2) ).

% Range_snd
tff(fact_7800_snd__eq__Range,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),image2(product_prod(B,A),A,product_snd(B,A)),R) = aa(set(product_prod(B,A)),set(A),range(B,A),R) ).

% snd_eq_Range
tff(fact_7801_finite__Range,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R2)
     => aa(set(B),$o,finite_finite2(B),aa(set(product_prod(A,B)),set(B),range(A,B),R2)) ) ).

% finite_Range
tff(fact_7802_Range__Un__eq,axiom,
    ! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B2: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(A),range(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),sup_sup(set(product_prod(B,A))),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(B,A)),set(A),range(B,A),A4)),aa(set(product_prod(B,A)),set(A),range(B,A),B2)) ).

% Range_Un_eq
tff(fact_7803_subset_Omaxchain__def,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A))] :
      ( pred_maxchain(set(A),A4,ord_less(set(A)),C2)
    <=> ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2)
        & ~ ? [S10: set(set(A))] :
              ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),S10)
              & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),C2),S10) ) ) ) ).

% subset.maxchain_def
tff(fact_7804_subset_Omaxchain__imp__chain,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A))] :
      ( pred_maxchain(set(A),A4,ord_less(set(A)),C2)
     => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2) ) ).

% subset.maxchain_imp_chain
tff(fact_7805_Range__Int__subset,axiom,
    ! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B2: set(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(B,A)),set(A),range(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),set(product_prod(B,A))),inf_inf(set(product_prod(B,A))),A4),B2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(B,A)),set(A),range(B,A),A4)),aa(set(product_prod(B,A)),set(A),range(B,A),B2))) ).

% Range_Int_subset
tff(fact_7806_Range__Diff__subset,axiom,
    ! [A: $tType,B: $tType,A4: set(product_prod(B,A)),B2: set(product_prod(B,A))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),aa(set(product_prod(B,A)),set(A),range(B,A),A4)),aa(set(product_prod(B,A)),set(A),range(B,A),B2))),aa(set(product_prod(B,A)),set(A),range(B,A),aa(set(product_prod(B,A)),set(product_prod(B,A)),minus_minus(set(product_prod(B,A)),A4),B2))) ).

% Range_Diff_subset
tff(fact_7807_subset_Onot__maxchain__Some,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2)
     => ( ~ pred_maxchain(set(A),A4,ord_less(set(A)),C2)
       => ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_ake(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C2)))
          & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),C2),fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_ake(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C2))) ) ) ) ).

% subset.not_maxchain_Some
tff(fact_7808_pred__on_Omaxchain__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C2: set(A)] :
      ( pred_maxchain(A,A4,P,C2)
    <=> ( aa(set(A),$o,pred_chain(A,A4,P),C2)
        & ~ ? [S10: set(A)] :
              ( aa(set(A),$o,pred_chain(A,A4,P),S10)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),C2),S10) ) ) ) ).

% pred_on.maxchain_def
tff(fact_7809_subset__maxchain__max,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A)),X4: set(A)] :
      ( pred_maxchain(set(A),A4,ord_less(set(A)),C2)
     => ( member(set(A),X4,A4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2)),X4)
         => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C2) = X4 ) ) ) ) ).

% subset_maxchain_max
tff(fact_7810_wf__UN,axiom,
    ! [B: $tType,A: $tType,I5: set(A),R2: fun(A,set(product_prod(B,B)))] :
      ( ! [I2: A] :
          ( member(A,I2,I5)
         => wf(B,aa(A,set(product_prod(B,B)),R2,I2)) )
     => ( ! [I2: A,J2: A] :
            ( member(A,I2,I5)
           => ( member(A,J2,I5)
             => ( ( aa(A,set(product_prod(B,B)),R2,I2) != aa(A,set(product_prod(B,B)),R2,J2) )
               => ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(product_prod(B,B)),set(B),domain(B,B),aa(A,set(product_prod(B,B)),R2,I2))),aa(set(product_prod(B,B)),set(B),range(B,B),aa(A,set(product_prod(B,B)),R2,J2))) = bot_bot(set(B)) ) ) ) )
       => wf(B,aa(set(set(product_prod(B,B))),set(product_prod(B,B)),complete_Sup_Sup(set(product_prod(B,B))),aa(set(A),set(set(product_prod(B,B))),image2(A,set(product_prod(B,B)),R2),I5))) ) ) ).

% wf_UN
tff(fact_7811_pred__on_Osuc__def,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),C2: set(A)] :
      pred_suc(A,A4,P,C2) = $ite(
        ( ~ aa(set(A),$o,pred_chain(A,A4,P),C2)
        | pred_maxchain(A,A4,P,C2) ),
        C2,
        fChoice(set(A),aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_akc(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),A4),P),C2)) ) ).

% pred_on.suc_def
tff(fact_7812_subset_Onot__chain__suc,axiom,
    ! [A: $tType,A4: set(set(A)),X4: set(set(A))] :
      ( ~ aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),X4)
     => ( pred_suc(set(A),A4,ord_less(set(A)),X4) = X4 ) ) ).

% subset.not_chain_suc
tff(fact_7813_Domain__Id__on,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(product_prod(A,A)),set(A),domain(A,A),id_on(A,A4)) = A4 ).

% Domain_Id_on
tff(fact_7814_subset_Omaxchain__suc,axiom,
    ! [A: $tType,A4: set(set(A)),X4: set(set(A))] :
      ( pred_maxchain(set(A),A4,ord_less(set(A)),X4)
     => ( pred_suc(set(A),A4,ord_less(set(A)),X4) = X4 ) ) ).

% subset.maxchain_suc
tff(fact_7815_Domain__empty,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(A,B)),set(A),domain(A,B),bot_bot(set(product_prod(A,B)))) = bot_bot(set(A)) ).

% Domain_empty
tff(fact_7816_Domain__Id,axiom,
    ! [A: $tType] : aa(set(product_prod(A,A)),set(A),domain(A,A),id2(A)) = top_top(set(A)) ).

% Domain_Id
tff(fact_7817_Range__converse,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : aa(set(product_prod(B,A)),set(A),range(B,A),converse(A,B,R2)) = aa(set(product_prod(A,B)),set(A),domain(A,B),R2) ).

% Range_converse
tff(fact_7818_Domain__converse,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A))] : aa(set(product_prod(A,B)),set(A),domain(A,B),converse(B,A,R2)) = aa(set(product_prod(B,A)),set(A),range(B,A),R2) ).

% Domain_converse
tff(fact_7819_Domain__Collect__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),P))) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_zm(fun(A,fun(B,$o)),fun(A,$o),P)) ).

% Domain_Collect_case_prod
tff(fact_7820_Domain__insert,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3)),R2)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A3),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ).

% Domain_insert
tff(fact_7821_subset_Osuc__def,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A))] :
      pred_suc(set(A),A4,ord_less(set(A)),C2) = $ite(
        ( ~ aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2)
        | pred_maxchain(set(A),A4,ord_less(set(A)),C2) ),
        C2,
        fChoice(set(set(A)),aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_ake(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),A4),C2)) ) ).

% subset.suc_def
tff(fact_7822_subset_Osuc__not__equals,axiom,
    ! [A: $tType,A4: set(set(A)),C2: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),C2)
     => ( ~ pred_maxchain(set(A),A4,ord_less(set(A)),C2)
       => ( pred_suc(set(A),A4,ord_less(set(A)),C2) != C2 ) ) ) ).

% subset.suc_not_equals
tff(fact_7823_pred__on_Ochain__sucD,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,fun(A,$o)),X4: set(A)] :
      ( aa(set(A),$o,pred_chain(A,A4,P),X4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),pred_suc(A,A4,P,X4)),A4)
        & aa(set(A),$o,pred_chain(A,A4,P),pred_suc(A,A4,P,X4)) ) ) ).

% pred_on.chain_sucD
tff(fact_7824_subset_Ochain__suc,axiom,
    ! [A: $tType,A4: set(set(A)),X4: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),X4)
     => aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),pred_suc(set(A),A4,ord_less(set(A)),X4)) ) ).

% subset.chain_suc
tff(fact_7825_Domain__Un__eq,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),sup_sup(set(product_prod(A,B))),A4),B2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),A4)),aa(set(product_prod(A,B)),set(A),domain(A,B),B2)) ).

% Domain_Un_eq
tff(fact_7826_finite__Domain,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,finite_finite2(product_prod(A,B)),R2)
     => aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).

% finite_Domain
tff(fact_7827_Domain__fst,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),R2) ).

% Domain_fst
tff(fact_7828_fst__eq__Domain,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),image2(product_prod(A,B),A,product_fst(A,B)),R) = aa(set(product_prod(A,B)),set(A),domain(A,B),R) ).

% fst_eq_Domain
tff(fact_7829_Domain__mono,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R2),S3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),R2)),aa(set(product_prod(A,B)),set(A),domain(A,B),S3)) ) ).

% Domain_mono
tff(fact_7830_pred__on_Osuc__in__carrier,axiom,
    ! [A: $tType,X4: set(A),A4: set(A),P: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),A4)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),pred_suc(A,A4,P,X4)),A4) ) ).

% pred_on.suc_in_carrier
tff(fact_7831_pred__on_Osuc__subset,axiom,
    ! [A: $tType,X4: set(A),A4: set(A),P: fun(A,fun(A,$o))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),pred_suc(A,A4,P,X4)) ).

% pred_on.suc_subset
tff(fact_7832_pred__on_Osubset__suc,axiom,
    ! [A: $tType,X4: set(A),Y3: set(A),A4: set(A),P: fun(A,fun(A,$o))] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),Y3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),pred_suc(A,A4,P,Y3)) ) ).

% pred_on.subset_suc
tff(fact_7833_subset_Osuc__in__carrier,axiom,
    ! [A: $tType,X4: set(set(A)),A4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X4),A4)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pred_suc(set(A),A4,ord_less(set(A)),X4)),A4) ) ).

% subset.suc_in_carrier
tff(fact_7834_subset_Osuc__subset,axiom,
    ! [A: $tType,X4: set(set(A)),A4: set(set(A))] : aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X4),pred_suc(set(A),A4,ord_less(set(A)),X4)) ).

% subset.suc_subset
tff(fact_7835_subset_Osubset__suc,axiom,
    ! [A: $tType,X4: set(set(A)),Y3: set(set(A)),A4: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X4),Y3)
     => aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),X4),pred_suc(set(A),A4,ord_less(set(A)),Y3)) ) ).

% subset.subset_suc
tff(fact_7836_Domain__empty__iff,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] :
      ( ( aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = bot_bot(set(A)) )
    <=> ( R2 = bot_bot(set(product_prod(A,B))) ) ) ).

% Domain_empty_iff
tff(fact_7837_Domain__iff,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
    <=> ? [Y4: B] : member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),Y4),R2) ) ).

% Domain_iff
tff(fact_7838_DomainE,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
     => ~ ! [B5: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B5),R2) ) ).

% DomainE
tff(fact_7839_Domain_ODomainI,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,R2: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B3),R2)
     => member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2)) ) ).

% Domain.DomainI
tff(fact_7840_Domain_Osimps,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
    <=> ? [A9: A,B7: B] :
          ( ( A3 = A9 )
          & member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A9),B7),R2) ) ) ).

% Domain.simps
tff(fact_7841_Domain_Ocases,axiom,
    ! [B: $tType,A: $tType,A3: A,R2: set(product_prod(A,B))] :
      ( member(A,A3,aa(set(product_prod(A,B)),set(A),domain(A,B),R2))
     => ~ ! [B5: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,A3),B5),R2) ) ).

% Domain.cases
tff(fact_7842_Domain__unfold,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(A),domain(A,B),R2) = aa(fun(A,$o),set(A),collect(A),aTP_Lamp_akf(set(product_prod(A,B)),fun(A,$o),R2)) ).

% Domain_unfold
tff(fact_7843_subset_Ochain__sucD,axiom,
    ! [A: $tType,A4: set(set(A)),X4: set(set(A))] :
      ( aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),X4)
     => ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),pred_suc(set(A),A4,ord_less(set(A)),X4)),A4)
        & aa(set(set(A)),$o,pred_chain(set(A),A4,ord_less(set(A))),pred_suc(set(A),A4,ord_less(set(A)),X4)) ) ) ).

% subset.chain_sucD
tff(fact_7844_Domain__Int__subset,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B2: set(product_prod(A,B))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),set(product_prod(A,B))),inf_inf(set(product_prod(A,B))),A4),B2))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,B)),set(A),domain(A,B),A4)),aa(set(product_prod(A,B)),set(A),domain(A,B),B2))) ).

% Domain_Int_subset
tff(fact_7845_Field__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : aa(set(product_prod(A,A)),set(A),field2(A),R2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R2)),aa(set(product_prod(A,A)),set(A),range(A,A),R2)) ).

% Field_def
tff(fact_7846_Domain__Diff__subset,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B2: set(product_prod(A,B))] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),aa(set(product_prod(A,B)),set(A),domain(A,B),A4)),aa(set(product_prod(A,B)),set(A),domain(A,B),B2))),aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(product_prod(A,B)),set(product_prod(A,B)),minus_minus(set(product_prod(A,B)),A4),B2))) ).

% Domain_Diff_subset
tff(fact_7847_Domain__Union,axiom,
    ! [B: $tType,A: $tType,S: set(set(product_prod(A,B)))] : aa(set(product_prod(A,B)),set(A),domain(A,B),aa(set(set(product_prod(A,B))),set(product_prod(A,B)),complete_Sup_Sup(set(product_prod(A,B))),S)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(product_prod(A,B))),set(set(A)),image2(set(product_prod(A,B)),set(A),domain(A,B)),S)) ).

% Domain_Union
tff(fact_7848_wf__Un,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),S3: set(product_prod(A,A))] :
      ( wf(A,R2)
     => ( wf(A,S3)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),R2)),aa(set(product_prod(A,A)),set(A),range(A,A),S3)) = bot_bot(set(A)) )
         => wf(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R2),S3)) ) ) ) ).

% wf_Un
tff(fact_7849_wf__Union,axiom,
    ! [A: $tType,R: set(set(product_prod(A,A)))] :
      ( ! [X3: set(product_prod(A,A))] :
          ( member(set(product_prod(A,A)),X3,R)
         => wf(A,X3) )
     => ( ! [X3: set(product_prod(A,A))] :
            ( member(set(product_prod(A,A)),X3,R)
           => ! [Xa4: set(product_prod(A,A))] :
                ( member(set(product_prod(A,A)),Xa4,R)
               => ( ( X3 != Xa4 )
                 => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(product_prod(A,A)),set(A),domain(A,A),X3)),aa(set(product_prod(A,A)),set(A),range(A,A),Xa4)) = bot_bot(set(A)) ) ) ) )
       => wf(A,aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),R)) ) ) ).

% wf_Union
tff(fact_7850_prod__set__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,Xa: B,Ya: A] : basic_snds(B,A,aa(A,product_prod(B,A),product_Pair(B,A,Xa),Ya)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Ya),bot_bot(set(A))) ).

% prod_set_simps(2)
tff(fact_7851_prod__set__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,Xa: A,Ya: B] : basic_fsts(A,B,aa(B,product_prod(A,B),product_Pair(A,B,Xa),Ya)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))) ).

% prod_set_simps(1)
tff(fact_7852_prod__set__defs_I1_J,axiom,
    ! [B: $tType,A: $tType,X2: product_prod(A,B)] : basic_fsts(A,B,X2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(product_prod(A,B),A,product_fst(A,B),X2)),bot_bot(set(A))) ).

% prod_set_defs(1)
tff(fact_7853_prod__set__defs_I2_J,axiom,
    ! [A: $tType,B: $tType,X2: product_prod(A,B)] : basic_snds(A,B,X2) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),aa(product_prod(A,B),B,product_snd(A,B),X2)),bot_bot(set(B))) ).

% prod_set_defs(2)
tff(fact_7854_iless__Suc__eq,axiom,
    ! [M: nat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_enat2(M)),extended_eSuc(Na))
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_enat2(M)),Na) ) ).

% iless_Suc_eq
tff(fact_7855_eSuc__mono,axiom,
    ! [Na: extended_enat,M: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),extended_eSuc(Na)),extended_eSuc(M))
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),M) ) ).

% eSuc_mono
tff(fact_7856_eSuc__ile__mono,axiom,
    ! [Na: extended_enat,M: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_eSuc(Na)),extended_eSuc(M))
    <=> aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Na),M) ) ).

% eSuc_ile_mono
tff(fact_7857_iless__eSuc0,axiom,
    ! [Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),Na),extended_eSuc(zero_zero(extended_enat)))
    <=> ( Na = zero_zero(extended_enat) ) ) ).

% iless_eSuc0
tff(fact_7858_sup__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(fun(filter(B),fun(filter(B),filter(B))),$o,aa(fun(filter(A),fun(filter(A),filter(A))),fun(fun(filter(B),fun(filter(B),filter(B))),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),filter(A)),fun(filter(B),filter(B)),rel_filter(A,B,A4),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,A4),rel_filter(A,B,A4))),sup_sup(filter(A))),sup_sup(filter(B))) ).

% sup_filter_parametric
tff(fact_7859_top__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_total(A,B,A4)
     => aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,A4),top_top(filter(A))),top_top(filter(B))) ) ).

% top_filter_parametric
tff(fact_7860_rel__filter__mono,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,$o)),B2: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A4),B2)
     => aa(fun(filter(A),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(B),$o)),fun(fun(filter(A),fun(filter(B),$o)),$o),ord_less_eq(fun(filter(A),fun(filter(B),$o))),rel_filter(A,B,A4)),rel_filter(A,B,B2)) ) ).

% rel_filter_mono
tff(fact_7861_ile__eSuc,axiom,
    ! [Na: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Na),extended_eSuc(Na)) ).

% ile_eSuc
tff(fact_7862_rel__filter__eq,axiom,
    ! [A: $tType] : rel_filter(A,A,fequal(A)) = fequal(filter(A)) ).

% rel_filter_eq
tff(fact_7863_bi__total__rel__filter,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_total(A,B,A4)
     => bi_total(filter(A),filter(B),rel_filter(A,B,A4)) ) ).

% bi_total_rel_filter
tff(fact_7864_eventually__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(filter(B),$o)),$o,aa(fun(fun(A,$o),fun(filter(A),$o)),fun(fun(fun(B,$o),fun(filter(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(filter(A),$o),fun(filter(B),$o),bNF_rel_fun(A,B,$o,$o,A4,fequal($o)),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,A4),fequal($o))),eventually(A)),eventually(B)) ).

% eventually_parametric
tff(fact_7865_filtermap__parametric,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,A4: fun(A,fun(C,$o)),B2: fun(B,fun(D,$o))] : aa(fun(fun(C,D),fun(filter(C),filter(D))),$o,aa(fun(fun(A,B),fun(filter(A),filter(B))),fun(fun(fun(C,D),fun(filter(C),filter(D))),$o),bNF_rel_fun(fun(A,B),fun(C,D),fun(filter(A),filter(B)),fun(filter(C),filter(D)),bNF_rel_fun(A,C,B,D,A4,B2),bNF_rel_fun(filter(A),filter(C),filter(B),filter(D),rel_filter(A,C,A4),rel_filter(B,D,B2))),filtermap(A,B)),filtermap(C,D)) ).

% filtermap_parametric
tff(fact_7866_bot__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,A4),bot_bot(filter(A))),bot_bot(filter(B))) ).

% bot_filter_parametric
tff(fact_7867_not__eSuc__ilei0,axiom,
    ! [Na: extended_enat] : ~ aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_eSuc(Na)),zero_zero(extended_enat)) ).

% not_eSuc_ilei0
tff(fact_7868_i0__iless__eSuc,axiom,
    ! [Na: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),zero_zero(extended_enat)),extended_eSuc(Na)) ).

% i0_iless_eSuc
tff(fact_7869_ileI1,axiom,
    ! [M: extended_enat,Na: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less(extended_enat),M),Na)
     => aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),extended_eSuc(M)),Na) ) ).

% ileI1
tff(fact_7870_rel__filter_Ocases,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),F3: filter(A),G2: filter(B)] :
      ( aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),F3),G2)
     => ~ ! [Z8: filter(product_prod(A,B))] :
            ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R)),Z8)
           => ( ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_fst(A,B)),Z8) = F3 )
             => ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_snd(A,B)),Z8) != G2 ) ) ) ) ).

% rel_filter.cases
tff(fact_7871_rel__filter_Osimps,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),F3: filter(A),G2: filter(B)] :
      ( aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),F3),G2)
    <=> ? [Z7: filter(product_prod(A,B))] :
          ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R)),Z7)
          & ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_fst(A,B)),Z7) = F3 )
          & ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_snd(A,B)),Z7) = G2 ) ) ) ).

% rel_filter.simps
tff(fact_7872_rel__filter_Ointros,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,$o)),Z6: filter(product_prod(A,B)),F3: filter(A),G2: filter(B)] :
      ( aa(filter(product_prod(A,B)),$o,aa(fun(product_prod(A,B),$o),fun(filter(product_prod(A,B)),$o),eventually(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R)),Z6)
     => ( ( aa(filter(product_prod(A,B)),filter(A),aa(fun(product_prod(A,B),A),fun(filter(product_prod(A,B)),filter(A)),map_filter_on(product_prod(A,B),A,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_fst(A,B)),Z6) = F3 )
       => ( ( aa(filter(product_prod(A,B)),filter(B),aa(fun(product_prod(A,B),B),fun(filter(product_prod(A,B)),filter(B)),map_filter_on(product_prod(A,B),B,aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),R))),product_snd(A,B)),Z6) = G2 )
         => aa(filter(B),$o,aa(filter(A),fun(filter(B),$o),rel_filter(A,B,R),F3),G2) ) ) ) ).

% rel_filter.intros
tff(fact_7873_subset__code_I3_J,axiom,
    ! [A: $tType] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),coset(A,nil(A))),aa(list(A),set(A),set2(A),nil(A))) ).

% subset_code(3)
tff(fact_7874_wo__rel_Oofilter__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( order_ofilter(A,R2,A4)
      <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(product_prod(A,A)),set(A),field2(A),R2))
          & ! [X: A] :
              ( member(A,X,A4)
             => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_under(A,R2,X)),A4) ) ) ) ) ).

% wo_rel.ofilter_def
tff(fact_7875_wo__rel_Oofilter__linord,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
            | aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4) ) ) ) ) ).

% wo_rel.ofilter_linord
tff(fact_7876_subset__code_I2_J,axiom,
    ! [A: $tType,A4: set(A),Ys2: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),coset(A,Ys2))
    <=> ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Ys2))
         => ~ member(A,X,A4) ) ) ).

% subset_code(2)
tff(fact_7877_ofilter__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_ofilter(A,R2,A4)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(product_prod(A,A)),set(A),field2(A),R2))
        & ! [X: A] :
            ( member(A,X,A4)
           => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_under(A,R2,X)),A4) ) ) ) ).

% ofilter_def
tff(fact_7878_insert__code_I2_J,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),coset(A,Xs)) = coset(A,removeAll(A,Xa,Xs)) ).

% insert_code(2)
tff(fact_7879_union__coset__filter,axiom,
    ! [A: $tType,Xs: list(A),A4: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),coset(A,Xs)),A4) = coset(A,filter2(A,aTP_Lamp_ba(set(A),fun(A,$o),A4),Xs)) ).

% union_coset_filter
tff(fact_7880_ofilterIncl__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : bNF_We413866401316099525erIncl(A,R2) = aa(fun(product_prod(set(A),set(A)),$o),set(product_prod(set(A),set(A))),collect(product_prod(set(A),set(A))),aa(fun(set(A),fun(set(A),$o)),fun(product_prod(set(A),set(A)),$o),product_case_prod(set(A),set(A),$o),aTP_Lamp_akg(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),R2))) ).

% ofilterIncl_def
tff(fact_7881_bsqr__ofilter,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),D3: set(product_prod(A,A))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(product_prod(A,A),bNF_Wellorder_bsqr(A,R2),D3)
       => ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less(set(product_prod(A,A))),D3),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_agh(set(product_prod(A,A)),fun(A,set(A)),R2)))
         => ( ~ ? [A5: A] : aa(set(product_prod(A,A)),set(A),field2(A),R2) = order_under(A,R2,A5)
           => ? [A7: set(A)] :
                ( order_ofilter(A,R2,A7)
                & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A7),aa(set(product_prod(A,A)),set(A),field2(A),R2))
                & aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),D3),product_Sigma(A,A,A7,aTP_Lamp_afz(set(A),fun(A,set(A)),A7))) ) ) ) ) ) ).

% bsqr_ofilter
tff(fact_7882_natLeq__on__well__order__on,axiom,
    ! [Na: nat] : order_well_order_on(nat,aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Na)),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Na)))) ).

% natLeq_on_well_order_on
tff(fact_7883_well__order__on__empty,axiom,
    ! [A: $tType] : order_well_order_on(A,bot_bot(set(A)),bot_bot(set(product_prod(A,A)))) ).

% well_order_on_empty
tff(fact_7884_well__order__on__Restr,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => order_well_order_on(A,A4,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))) ) ) ).

% well_order_on_Restr
tff(fact_7885_natLeq__on__Well__order,axiom,
    ! [Na: nat] : order_well_order_on(nat,aa(set(product_prod(nat,nat)),set(nat),field2(nat),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Na)))),aa(fun(product_prod(nat,nat),$o),set(product_prod(nat,nat)),collect(product_prod(nat,nat)),aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Na)))) ).

% natLeq_on_Well_order
tff(fact_7886_Linear__order__Well__order__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( order_679001287576687338der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
      <=> ! [A8: set(A)] :
            ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),aa(set(product_prod(A,A)),set(A),field2(A),R2))
           => ( ( A8 != bot_bot(set(A)) )
             => ? [X: A] :
                  ( member(A,X,A8)
                  & ! [Xa3: A] :
                      ( member(A,Xa3,A8)
                     => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Xa3),R2) ) ) ) ) ) ) ).

% Linear_order_Well_order_iff
tff(fact_7887_ofilter__Restr__subset,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
         => order_ofilter(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,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2))),A4) ) ) ) ).

% ofilter_Restr_subset
tff(fact_7888_ofilter__subset__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
          <=> 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))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2)))),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ) ).

% ofilter_subset_ordLess
tff(fact_7889_ofilter__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),aa(set(product_prod(A,A)),set(A),field2(A),R2))
        <=> 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))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))),R2),bNF_We4044943003108391690rdLess(A,A)) ) ) ) ).

% ofilter_ordLess
tff(fact_7890_finite__ordLess__infinite,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B))] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_well_order_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R4),R4)
       => ( aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ( ~ aa(set(B),$o,finite_finite2(B),aa(set(product_prod(B,B)),set(B),field2(B),R4))
           => member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R2),R4),bNF_We4044943003108391690rdLess(A,B)) ) ) ) ) ).

% finite_ordLess_infinite
tff(fact_7891_underS__Restr__ordLess,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
       => 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))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,order_underS(A,R2,A3),aa(A,fun(A,set(A)),aTP_Lamp_akh(set(product_prod(A,A)),fun(A,fun(A,set(A))),R2),A3)))),R2),bNF_We4044943003108391690rdLess(A,A)) ) ) ).

% underS_Restr_ordLess
tff(fact_7892_ofilter__subset__ordLeq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
          <=> 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))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2)))),bNF_Wellorder_ordLeq(A,A)) ) ) ) ) ).

% ofilter_subset_ordLeq
tff(fact_7893_wo__rel_Oofilter__AboveS__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( order_ofilter(A,R2,A4)
       => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),order_AboveS(A,R2,A4)) = aa(set(product_prod(A,A)),set(A),field2(A),R2) ) ) ) ).

% wo_rel.ofilter_AboveS_Field
tff(fact_7894_AboveS__Field,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),order_AboveS(A,R2,A4)),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ).

% AboveS_Field
tff(fact_7895_AboveS__disjoint,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A))] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A4),order_AboveS(A,R2,A4)) = bot_bot(set(A)) ).

% AboveS_disjoint
tff(fact_7896_wo__rel_Osuc__greater,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A),B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( order_AboveS(A,R2,B2) != bot_bot(set(A)) )
         => ( member(A,B3,B2)
           => ( ( bNF_Wellorder_wo_suc(A,R2,B2) != B3 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),bNF_Wellorder_wo_suc(A,R2,B2)),R2) ) ) ) ) ) ).

% wo_rel.suc_greater
tff(fact_7897_wo__rel_Osuc__ofilter__in,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( order_ofilter(A,R2,A4)
       => ( ( order_AboveS(A,R2,A4) != bot_bot(set(A)) )
         => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B3),bNF_Wellorder_wo_suc(A,R2,A4)),R2)
           => ( ( B3 != bNF_Wellorder_wo_suc(A,R2,A4) )
             => member(A,B3,A4) ) ) ) ) ) ).

% wo_rel.suc_ofilter_in
tff(fact_7898_wo__rel_Oequals__suc__AboveS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A),A3: A] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,A3,order_AboveS(A,R2,B2))
         => ( ! [A20: A] :
                ( member(A,A20,order_AboveS(A,R2,B2))
               => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),A20),R2) )
           => ( A3 = bNF_Wellorder_wo_suc(A,R2,B2) ) ) ) ) ) ).

% wo_rel.equals_suc_AboveS
tff(fact_7899_wo__rel_Osuc__inField,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( order_AboveS(A,R2,B2) != bot_bot(set(A)) )
         => member(A,bNF_Wellorder_wo_suc(A,R2,B2),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ) ) ).

% wo_rel.suc_inField
tff(fact_7900_wo__rel_Osuc__AboveS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),B2: set(A)] :
      ( bNF_Wellorder_wo_rel(A,R2)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( order_AboveS(A,R2,B2) != bot_bot(set(A)) )
         => member(A,bNF_Wellorder_wo_suc(A,R2,B2),order_AboveS(A,R2,B2)) ) ) ) ).

% wo_rel.suc_AboveS
tff(fact_7901_inf__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => ( bi_total(A,B,A4)
       => aa(fun(filter(B),fun(filter(B),filter(B))),$o,aa(fun(filter(A),fun(filter(A),filter(A))),fun(fun(filter(B),fun(filter(B),filter(B))),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),filter(A)),fun(filter(B),filter(B)),rel_filter(A,B,A4),bNF_rel_fun(filter(A),filter(B),filter(A),filter(B),rel_filter(A,B,A4),rel_filter(A,B,A4))),inf_inf(filter(A))),inf_inf(filter(B))) ) ) ).

% inf_filter_parametric
tff(fact_7902_frequently__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] : aa(fun(fun(B,$o),fun(filter(B),$o)),$o,aa(fun(fun(A,$o),fun(filter(A),$o)),fun(fun(fun(B,$o),fun(filter(B),$o)),$o),bNF_rel_fun(fun(A,$o),fun(B,$o),fun(filter(A),$o),fun(filter(B),$o),bNF_rel_fun(A,B,$o,$o,A4,fequal($o)),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,A4),fequal($o))),frequently(A)),frequently(B)) ).

% frequently_parametric
tff(fact_7903_frequently__const,axiom,
    ! [A: $tType,F3: filter(A),P: $o] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_mb($o,fun(A,$o),(P))),F3)
      <=> (P) ) ) ).

% frequently_const
tff(fact_7904_frequently__bex__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),F3: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aef(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P)),F3)
      <=> ? [X: A] :
            ( member(A,X,A4)
            & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X)),F3) ) ) ) ).

% frequently_bex_finite_distrib
tff(fact_7905_frequently__bex__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),F3: filter(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aef(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P)),F3)
       => ? [X3: A] :
            ( member(A,X3,A4)
            & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X3)),F3) ) ) ) ).

% frequently_bex_finite
tff(fact_7906_frequently__all,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_aki(fun(A,fun(B,$o)),fun(A,$o),P)),F3)
    <=> ! [Y5: fun(A,B)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_zn(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y5)),F3) ) ).

% frequently_all
tff(fact_7907_not__frequently__False,axiom,
    ! [A: $tType,F3: filter(A)] : ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_ao(A,$o)),F3) ).

% not_frequently_False
tff(fact_7908_frequently__disj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
        | aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F3) ) ) ).

% frequently_disj_iff
tff(fact_7909_frequently__elim1,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
     => ( ! [I2: A] :
            ( aa(A,$o,P,I2)
           => aa(A,$o,Q,I2) )
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F3) ) ) ).

% frequently_elim1
tff(fact_7910_frequently__disj,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3) ) ) ).

% frequently_disj
tff(fact_7911_frequently__ex,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% frequently_ex
tff(fact_7912_eventually__frequently__const__simps_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),C2: $o,F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa($o,fun(A,$o),aTP_Lamp_akj(fun(A,$o),fun($o,fun(A,$o)),P),(C2))),F3)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
        & (C2) ) ) ).

% eventually_frequently_const_simps(1)
tff(fact_7913_eventually__frequently__const__simps_I2_J,axiom,
    ! [A: $tType,C2: $o,P: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_akk($o,fun(fun(A,$o),fun(A,$o)),(C2)),P)),F3)
    <=> ( (C2)
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3) ) ) ).

% eventually_frequently_const_simps(2)
tff(fact_7914_frequently__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( ! [X3: A] :
          ( aa(A,$o,P,X3)
         => aa(A,$o,Q,X3) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F3) ) ) ).

% frequently_mono
tff(fact_7915_frequentlyE,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
     => ~ ! [X3: A] : ~ aa(A,$o,P,X3) ) ).

% frequentlyE
tff(fact_7916_eventually__frequently__const__simps_I5_J,axiom,
    ! [A: $tType,P: fun(A,$o),C2: $o,F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa($o,fun(A,$o),aTP_Lamp_akl(fun(A,$o),fun($o,fun(A,$o)),P),(C2))),F3)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
       => (C2) ) ) ).

% eventually_frequently_const_simps(5)
tff(fact_7917_frequently__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F3) ) ) ).

% frequently_mp
tff(fact_7918_frequently__def,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
    <=> ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)),F3) ) ).

% frequently_def
tff(fact_7919_not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)),F3) ) ).

% not_eventually
tff(fact_7920_not__frequently,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A)] :
      ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)),F3) ) ).

% not_frequently
tff(fact_7921_frequently__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F3) ) ) ).

% frequently_rev_mp
tff(fact_7922_frequently__imp__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F3) ) ) ).

% frequently_imp_iff
tff(fact_7923_eventually__frequentlyE,axiom,
    ! [A: $tType,P: fun(A,$o),F3: filter(A),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_akm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),F3)
       => ( ( F3 != bot_bot(filter(A)) )
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),F3) ) ) ) ).

% eventually_frequentlyE
tff(fact_7924_eventually__frequently,axiom,
    ! [A: $tType,F3: filter(A),P: fun(A,$o)] :
      ( ( F3 != bot_bot(filter(A)) )
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),F3)
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),F3) ) ) ).

% eventually_frequently
tff(fact_7925_frequently__const__iff,axiom,
    ! [A: $tType,P: $o,F3: filter(A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_mb($o,fun(A,$o),(P))),F3)
    <=> ( (P)
        & ( F3 != bot_bot(filter(A)) ) ) ) ).

% frequently_const_iff
tff(fact_7926_bi__unique__rel__filter,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => bi_unique(filter(A),filter(B),rel_filter(A,B,A4)) ) ).

% bi_unique_rel_filter
tff(fact_7927_le__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => aa(fun(filter(B),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(A),$o)),fun(fun(filter(B),fun(filter(B),$o)),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),$o),fun(filter(B),$o),rel_filter(A,B,A4),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,A4),fequal($o))),ord_less_eq(filter(A))),ord_less_eq(filter(B))) ) ).

% le_filter_parametric
tff(fact_7928_frequently__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,$o),A3: A,S: set(A)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),topolo174197925503356063within(A,A3,S))
        <=> ! [D5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),D5)
             => ? [X: A] :
                  ( member(A,X,S)
                  & ( X != A3 )
                  & aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,X,A3)),D5)
                  & aa(A,$o,P,X) ) ) ) ) ).

% frequently_at
tff(fact_7929_less__filter__parametric,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,$o))] :
      ( bi_unique(A,B,A4)
     => aa(fun(filter(B),fun(filter(B),$o)),$o,aa(fun(filter(A),fun(filter(A),$o)),fun(fun(filter(B),fun(filter(B),$o)),$o),bNF_rel_fun(filter(A),filter(B),fun(filter(A),$o),fun(filter(B),$o),rel_filter(A,B,A4),bNF_rel_fun(filter(A),filter(B),$o,$o,rel_filter(A,B,A4),fequal($o))),ord_less(filter(A))),ord_less(filter(B))) ) ).

% less_filter_parametric
tff(fact_7930_card__of__UNION__ordLeq__infinite,axiom,
    ! [B: $tType,A: $tType,C: $tType,B2: set(A),I5: set(B),A4: fun(B,set(C))] :
      ( ~ aa(set(A),$o,finite_finite2(A),B2)
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,I5)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(B,A))
       => ( ! [X3: B] :
              ( member(B,X3,I5)
             => member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(C,A)) )
         => member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5)))),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(C,A)) ) ) ) ).

% card_of_UNION_ordLeq_infinite
tff(fact_7931_List_Oset__insert,axiom,
    ! [A: $tType,Xa: A,Xs: list(A)] : aa(list(A),set(A),set2(A),insert(A,Xa,Xs)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),aa(list(A),set(A),set2(A),Xs)) ).

% List.set_insert
tff(fact_7932_card__of__mono1,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
     => 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)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(A,A)) ) ).

% card_of_mono1
tff(fact_7933_ex__bij__betw,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),R2),bNF_Wellorder_ordLeq(A,B))
     => ? [F5: fun(B,A),B4: set(B)] : bij_betw(B,A,F5,B4,A4) ) ).

% ex_bij_betw
tff(fact_7934_card__of__Times2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B)))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(A,B),product_prod(A,B))),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordLeq(B,product_prod(A,B))) ) ).

% card_of_Times2
tff(fact_7935_card__of__Times1,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => member(product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A)))),product_Pair(set(product_prod(B,B)),set(product_prod(product_prod(B,A),product_prod(B,A))),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B2,aTP_Lamp_ld(set(A),fun(B,set(A)),A4)))),bNF_Wellorder_ordLeq(B,product_prod(B,A))) ) ).

% card_of_Times1
tff(fact_7936_card__of__empty,axiom,
    ! [B: $tType,A: $tType,A4: set(B)] : member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Wellorder_ordLeq(A,B)) ).

% card_of_empty
tff(fact_7937_card__of__empty3,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordLeq(A,B))
     => ( A4 = bot_bot(set(A)) ) ) ).

% card_of_empty3
tff(fact_7938_card__of__Sigma__ordLeq__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,B2: set(A),I5: set(B),A4: fun(B,set(C))] :
      ( ~ aa(set(A),$o,finite_finite2(A),B2)
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,I5)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(B,A))
       => ( ! [X3: B] :
              ( member(B,X3,I5)
             => member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(C,A)) )
         => member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I5,A4))),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ).

% card_of_Sigma_ordLeq_infinite
tff(fact_7939_infinite__iff__natLeq__ordLeq,axiom,
    ! [A: $tType,A4: set(A)] :
      ~ ( aa(set(A),$o,finite_finite2(A),A4)
      <=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A)),bNF_Ca8665028551170535155natLeq),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(nat,A)) ) ).

% infinite_iff_natLeq_ordLeq
tff(fact_7940_card__of__ordLeq__finite,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => aa(set(A),$o,finite_finite2(A),A4) ) ) ).

% card_of_ordLeq_finite
tff(fact_7941_card__of__ordLeq__infinite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ~ aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% card_of_ordLeq_infinite
tff(fact_7942_infinite__iff__card__of__nat,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
    <=> member(product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(nat,nat)),set(product_prod(A,A))),product_Pair(set(product_prod(nat,nat)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(nat,top_top(set(nat)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(nat,A)) ) ).

% infinite_iff_card_of_nat
tff(fact_7943_finite__iff__ordLess__natLeq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
    <=> member(product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),aa(set(product_prod(nat,nat)),product_prod(set(product_prod(A,A)),set(product_prod(nat,nat))),product_Pair(set(product_prod(A,A)),set(product_prod(nat,nat)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca8665028551170535155natLeq),bNF_We4044943003108391690rdLess(A,nat)) ) ).

% finite_iff_ordLess_natLeq
tff(fact_7944_card__of__ordLeq2,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( ( A4 != bot_bot(set(A)) )
     => ( ? [G6: fun(B,A)] : aa(set(B),set(A),image2(B,A,G6),B2) = A4
      <=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B)) ) ) ).

% card_of_ordLeq2
tff(fact_7945_surj__imp__ordLeq,axiom,
    ! [B: $tType,A: $tType,B2: set(A),F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),A4))
     => member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Wellorder_ordLeq(A,B)) ) ).

% surj_imp_ordLeq
tff(fact_7946_card__of__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B3: B] :
      ( ( A4 != bot_bot(set(A)) )
     => member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B3),bot_bot(set(B))))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A)) ) ).

% card_of_singl_ordLeq
tff(fact_7947_card__of__ordLess2,axiom,
    ! [A: $tType,B: $tType,B2: set(A),A4: set(B)] :
      ( ( B2 != bot_bot(set(A)) )
     => ( ~ ? [F7: fun(B,A)] : aa(set(B),set(A),image2(B,A,F7),A4) = B2
      <=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Ca6860139660246222851ard_of(A,B2)),bNF_We4044943003108391690rdLess(B,A)) ) ) ).

% card_of_ordLess2
tff(fact_7948_card__of__ordLeq,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( ? [F7: fun(A,B)] :
          ( inj_on(A,B,F7,A4)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A4)),B2) )
    <=> member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B)) ) ).

% card_of_ordLeq
tff(fact_7949_card__of__ordLess,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ~ ? [F7: fun(A,B)] :
            ( inj_on(A,B,F7,A4)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F7),A4)),B2) )
    <=> member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_We4044943003108391690rdLess(B,A)) ) ).

% card_of_ordLess
tff(fact_7950_ordLeq3__finite__infinite,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( ~ aa(set(B),$o,finite_finite2(B),B2)
       => member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B)) ) ) ).

% ordLeq3_finite_infinite
tff(fact_7951_card__of__Plus__Times__aux,axiom,
    ! [B: $tType,A: $tType,A12: A,A23: A,A4: set(A),B2: set(B)] :
      ( ( ( A12 != A23 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A23),bot_bot(set(A))))),A4) )
     => ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordLeq(A,B))
       => member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B2))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).

% card_of_Plus_Times_aux
tff(fact_7952_finite__Plus__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(sum_sum(A,B)),$o,finite_finite2(sum_sum(A,B)),sum_Plus(A,B,A4,B2))
    <=> ( aa(set(A),$o,finite_finite2(A),A4)
        & aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% finite_Plus_iff
tff(fact_7953_card__of__Un__Plus__ordLeq,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))),bNF_Ca6860139660246222851ard_of(sum_sum(A,A),sum_Plus(A,A,A4,B2))),bNF_Wellorder_ordLeq(A,sum_sum(A,A))) ).

% card_of_Un_Plus_ordLeq
tff(fact_7954_card__Plus,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => ( aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A4,B2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B2)) ) ) ) ).

% card_Plus
tff(fact_7955_finite__Plus,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(set(B),$o,finite_finite2(B),B2)
       => aa(set(sum_sum(A,B)),$o,finite_finite2(sum_sum(A,B)),sum_Plus(A,B,A4,B2)) ) ) ).

% finite_Plus
tff(fact_7956_finite__PlusD_I1_J,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(sum_sum(A,B)),$o,finite_finite2(sum_sum(A,B)),sum_Plus(A,B,A4,B2))
     => aa(set(A),$o,finite_finite2(A),A4) ) ).

% finite_PlusD(1)
tff(fact_7957_finite__PlusD_I2_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( aa(set(sum_sum(A,B)),$o,finite_finite2(sum_sum(A,B)),sum_Plus(A,B,A4,B2))
     => aa(set(B),$o,finite_finite2(B),B2) ) ).

% finite_PlusD(2)
tff(fact_7958_card__Plus__conv__if,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      aa(set(sum_sum(A,B)),nat,finite_card(sum_sum(A,B)),sum_Plus(A,B,A4,B2)) = $ite(
        ( aa(set(A),$o,finite_finite2(A),A4)
        & aa(set(B),$o,finite_finite2(B),B2) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,finite_card(A),A4)),aa(set(B),nat,finite_card(B),B2)),
        zero_zero(nat) ) ).

% card_Plus_conv_if
tff(fact_7959_card__of__Plus__ordLess__infinite,axiom,
    ! [A: $tType,C: $tType,B: $tType,C2: set(A),A4: set(B),B2: set(C)] :
      ( ~ aa(set(A),$o,finite_finite2(A),C2)
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Ca6860139660246222851ard_of(A,C2)),bNF_We4044943003108391690rdLess(B,A))
       => ( member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,B2)),bNF_Ca6860139660246222851ard_of(A,C2)),bNF_We4044943003108391690rdLess(C,A))
         => member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A4,B2))),bNF_Ca6860139660246222851ard_of(A,C2)),bNF_We4044943003108391690rdLess(sum_sum(B,C),A)) ) ) ) ).

% card_of_Plus_ordLess_infinite
tff(fact_7960_card__of__Plus__Times,axiom,
    ! [B: $tType,A: $tType,A12: A,A23: A,A4: set(A),B16: B,B23: B,B2: set(B)] :
      ( ( ( A12 != A23 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A23),bot_bot(set(A))))),A4) )
     => ( ( ( B16 != B23 )
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B16),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B23),bot_bot(set(B))))),B2) )
       => member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B)))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(product_prod(A,B),product_prod(A,B))),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B2))),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordLeq(sum_sum(A,B),product_prod(A,B))) ) ) ).

% card_of_Plus_Times
tff(fact_7961_Plus__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ( sum_Plus(A,B,A4,B2) = bot_bot(set(sum_sum(A,B))) )
    <=> ( ( A4 = bot_bot(set(A)) )
        & ( B2 = bot_bot(set(B)) ) ) ) ).

% Plus_eq_empty_conv
tff(fact_7962_card__of__Times__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B),B2: set(C)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,A4)),R2),bNF_Wellorder_ordLeq(B,A))
       => ( member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,B2)),R2),bNF_Wellorder_ordLeq(C,A))
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
           => member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,A4,aTP_Lamp_agz(set(C),fun(B,set(C)),B2)))),R2),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ) ).

% card_of_Times_ordLeq_infinite_Field
tff(fact_7963_Card__order__trans,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xa: A,Ya: A,Z: A] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( Xa != Ya )
       => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Ya),R2)
         => ( ( Ya != Z )
           => ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Ya),Z),R2)
             => ( ( Xa != Z )
                & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),Z),R2) ) ) ) ) ) ) ).

% Card_order_trans
tff(fact_7964_infinite__Card__order__limit,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A3: A] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( member(A,A3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
         => ? [X3: A] :
              ( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
              & ( A3 != X3 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A3),X3),R2) ) ) ) ) ).

% infinite_Card_order_limit
tff(fact_7965_Card__order__wo__rel,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => bNF_Wellorder_wo_rel(A,R2) ) ).

% Card_order_wo_rel
tff(fact_7966_Card__order__infinite__not__under,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ~ ? [A10: A] : aa(set(product_prod(A,A)),set(A),field2(A),R2) = order_under(A,R2,A10) ) ) ).

% Card_order_infinite_not_under
tff(fact_7967_Card__order__empty,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R2),bNF_Wellorder_ordLeq(B,A)) ) ).

% Card_order_empty
tff(fact_7968_Card__order__singl__ordLeq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),B3: B] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) )
       => member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),B3),bot_bot(set(B))))),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ).

% Card_order_singl_ordLeq
tff(fact_7969_card__of__Un__ordLeq__infinite__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,A4)),R2),bNF_Wellorder_ordLeq(B,A))
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),R2),bNF_Wellorder_ordLeq(B,A))
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
           => member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A4),B2))),R2),bNF_Wellorder_ordLeq(B,A)) ) ) ) ) ).

% card_of_Un_ordLeq_infinite_Field
tff(fact_7970_card__of__empty1,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A))] :
      ( ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
        | bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),R2),bNF_Wellorder_ordLeq(B,A)) ) ).

% card_of_empty1
tff(fact_7971_Card__order__Times2,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),A4: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( A4 != bot_bot(set(B)) )
       => member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))),R2),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,A4,aTP_Lamp_akn(set(product_prod(A,A)),fun(B,set(A)),R2)))),bNF_Wellorder_ordLeq(A,product_prod(B,A))) ) ) ).

% Card_order_Times2
tff(fact_7972_Card__order__Times1,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),B2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( B2 != bot_bot(set(B)) )
       => member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))),R2),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordLeq(A,product_prod(A,B))) ) ) ).

% Card_order_Times1
tff(fact_7973_Card__order__Times__same__infinite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_agh(set(product_prod(A,A)),fun(A,set(A)),R2)))),R2),bNF_Wellorder_ordLeq(product_prod(A,A),A)) ) ) ).

% Card_order_Times_same_infinite
tff(fact_7974_card__of__UNION__ordLeq__infinite__Field,axiom,
    ! [B: $tType,A: $tType,C: $tType,R2: set(product_prod(A,A)),I5: set(B),A4: fun(B,set(C))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,I5)),R2),bNF_Wellorder_ordLeq(B,A))
         => ( ! [X3: B] :
                ( member(B,X3,I5)
               => member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),R2),bNF_Wellorder_ordLeq(C,A)) )
           => member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),A4),I5)))),R2),bNF_Wellorder_ordLeq(C,A)) ) ) ) ) ).

% card_of_UNION_ordLeq_infinite_Field
tff(fact_7975_card__of__Plus__ordLess__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B),B2: set(C)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,A4)),R2),bNF_We4044943003108391690rdLess(B,A))
         => ( member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,B2)),R2),bNF_We4044943003108391690rdLess(C,A))
           => member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A4,B2))),R2),bNF_We4044943003108391690rdLess(sum_sum(B,C),A)) ) ) ) ) ).

% card_of_Plus_ordLess_infinite_Field
tff(fact_7976_card__of__Plus__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B),B2: set(C)] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,A4)),R2),bNF_Wellorder_ordLeq(B,A))
       => ( member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,B2)),R2),bNF_Wellorder_ordLeq(C,A))
         => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
           => member(product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(B,C),sum_sum(B,C))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(B,C),sum_Plus(B,C,A4,B2))),R2),bNF_Wellorder_ordLeq(sum_sum(B,C),A)) ) ) ) ) ).

% card_of_Plus_ordLeq_infinite_Field
tff(fact_7977_card__of__Sigma__ordLeq__infinite__Field,axiom,
    ! [A: $tType,C: $tType,B: $tType,R2: set(product_prod(A,A)),I5: set(B),A4: fun(B,set(C))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,I5)),R2),bNF_Wellorder_ordLeq(B,A))
         => ( ! [X3: B] :
                ( member(B,X3,I5)
               => member(product_prod(set(product_prod(C,C)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(C,C)),set(product_prod(A,A))),product_Pair(set(product_prod(C,C)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(C,aa(B,set(C),A4,X3))),R2),bNF_Wellorder_ordLeq(C,A)) )
           => member(product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(B,C),product_prod(B,C))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(B,C),product_Sigma(B,C,I5,A4))),R2),bNF_Wellorder_ordLeq(product_prod(B,C),A)) ) ) ) ) ).

% card_of_Sigma_ordLeq_infinite_Field
tff(fact_7978_regularCard__UNION,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As: fun(A,set(B)),B2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Ca7133664381575040944arCard(A,R2)
       => ( bNF_Ca3754400796208372196lChain(A,set(B),R2,As)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image2(A,set(B),As),aa(set(product_prod(A,A)),set(A),field2(A),R2))))
           => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),R2),bNF_We4044943003108391690rdLess(B,A))
             => ? [X3: A] :
                  ( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
                  & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(A,set(B),As,X3)) ) ) ) ) ) ) ).

% regularCard_UNION
tff(fact_7979_toCard__pred__def,axiom,
    ! [A: $tType,B: $tType,A4: set(A),R2: set(product_prod(B,B)),F2: fun(A,B)] :
      ( bNF_Gr1419584066657907630d_pred(A,B,A4,R2,F2)
    <=> ( inj_on(A,B,F2,A4)
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),A4)),aa(set(product_prod(B,B)),set(B),field2(B),R2))
        & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2) ) ) ).

% toCard_pred_def
tff(fact_7980_cardSuc__UNION,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As: fun(set(A),set(B)),B2: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R2),As)
         => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))))
           => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),R2),bNF_Wellorder_ordLeq(B,A))
             => ? [X3: set(A)] :
                  ( member(set(A),X3,aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))
                  & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),As,X3)) ) ) ) ) ) ) ).

% cardSuc_UNION
tff(fact_7981_Card__order__Times__infinite,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( ( aa(set(product_prod(B,B)),set(B),field2(B),P3) != bot_bot(set(B)) )
         => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),P3),R2),bNF_Wellorder_ordLeq(B,A))
           => ( member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R2),aTP_Lamp_ako(set(product_prod(B,B)),fun(A,set(B)),P3)))),R2),bNF_Wellorder_ordIso(product_prod(A,B),A))
              & member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,aa(set(product_prod(B,B)),set(B),field2(B),P3),aTP_Lamp_akn(set(product_prod(A,A)),fun(B,set(A)),R2)))),R2),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ) ).

% Card_order_Times_infinite
tff(fact_7982_finite__well__order__on__ordIso,axiom,
    ! [A: $tType,A4: set(A),R2: set(product_prod(A,A)),R4: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( order_well_order_on(A,A4,R2)
       => ( order_well_order_on(A,A4,R4)
         => 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)),R2),R4),bNF_Wellorder_ordIso(A,A)) ) ) ) ).

% finite_well_order_on_ordIso
tff(fact_7983_card__of__empty2,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B))
     => ( A4 = bot_bot(set(A)) ) ) ).

% card_of_empty2
tff(fact_7984_card__of__empty__ordIso,axiom,
    ! [B: $tType,A: $tType] : member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A)))),bNF_Ca6860139660246222851ard_of(B,bot_bot(set(B)))),bNF_Wellorder_ordIso(A,B)) ).

% card_of_empty_ordIso
tff(fact_7985_card__of__ordIso__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(A),$o,finite_finite2(A),A4)
      <=> aa(set(B),$o,finite_finite2(B),B2) ) ) ).

% card_of_ordIso_finite
tff(fact_7986_internalize__ordLeq,axiom,
    ! [A: $tType,B: $tType,R4: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R4),R2),bNF_Wellorder_ordLeq(A,B))
    <=> ? [P7: set(product_prod(B,B))] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(product_prod(B,B)),set(B),field2(B),P7)),aa(set(product_prod(B,B)),set(B),field2(B),R2))
          & member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R4),P7),bNF_Wellorder_ordIso(A,B))
          & member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B)),P7),R2),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_ordLeq
tff(fact_7987_infinite__cardSuc__regularCard,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => bNF_Ca7133664381575040944arCard(set(A),bNF_Ca8387033319878233205ardSuc(A,R2)) ) ) ).

% infinite_cardSuc_regularCard
tff(fact_7988_internalize__card__of__ordLeq2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),C2: set(B)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,C2)),bNF_Wellorder_ordLeq(A,B))
    <=> ? [B11: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),C2)
          & member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B11)),bNF_Wellorder_ordIso(A,B))
          & member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(B,B11)),bNF_Ca6860139660246222851ard_of(B,C2)),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_card_of_ordLeq2
tff(fact_7989_internalize__ordLess,axiom,
    ! [A: $tType,B: $tType,R4: set(product_prod(A,A)),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R4),R2),bNF_We4044943003108391690rdLess(A,B))
    <=> ? [P7: set(product_prod(B,B))] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(product_prod(B,B)),set(B),field2(B),P7)),aa(set(product_prod(B,B)),set(B),field2(B),R2))
          & member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R4),P7),bNF_Wellorder_ordIso(A,B))
          & member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B)),P7),R2),bNF_We4044943003108391690rdLess(B,B)) ) ) ).

% internalize_ordLess
tff(fact_7990_card__of__cardSuc__finite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( aa(set(set(A)),$o,finite_finite2(set(A)),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,bNF_Ca6860139660246222851ard_of(A,A4))))
    <=> aa(set(A),$o,finite_finite2(A),A4) ) ).

% card_of_cardSuc_finite
tff(fact_7991_internalize__card__of__ordLeq,axiom,
    ! [A: $tType,B: $tType,A4: set(A),R2: set(product_prod(B,B))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),R2),bNF_Wellorder_ordLeq(A,B))
    <=> ? [B11: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),aa(set(product_prod(B,B)),set(B),field2(B),R2))
          & member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(B,B11)),bNF_Wellorder_ordIso(A,B))
          & member(product_prod(set(product_prod(B,B)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(B,B)),set(product_prod(B,B))),product_Pair(set(product_prod(B,B)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(B,B11)),R2),bNF_Wellorder_ordLeq(B,B)) ) ) ).

% internalize_card_of_ordLeq
tff(fact_7992_card__of__ordIso__finite__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),A4: set(B)] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R2),bNF_Ca6860139660246222851ard_of(B,A4)),bNF_Wellorder_ordIso(A,B))
       => ( aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
        <=> aa(set(B),$o,finite_finite2(B),A4) ) ) ) ).

% card_of_ordIso_finite_Field
tff(fact_7993_cardSuc__finite,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( aa(set(set(A)),$o,finite_finite2(set(A)),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))
      <=> aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ) ).

% cardSuc_finite
tff(fact_7994_card__of__Times__same__infinite,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => member(product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(A,A),product_prod(A,A))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(A,A),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(A,A),A)) ) ).

% card_of_Times_same_infinite
tff(fact_7995_card__of__Times__infinite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B2 != bot_bot(set(B)) )
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))
         => ( member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(A,B),A))
            & member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B2,aTP_Lamp_ld(set(A),fun(B,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ) ).

% card_of_Times_infinite
tff(fact_7996_card__of__Times__infinite__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B2 != bot_bot(set(B)) )
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))
         => member(product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(A,B),product_prod(A,B))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(A,B),A)) ) ) ) ).

% card_of_Times_infinite_simps(1)
tff(fact_7997_card__of__Times__infinite__simps_I3_J,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B2 != bot_bot(set(B)) )
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))
         => member(product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A))),product_Pair(set(product_prod(product_prod(B,A),product_prod(B,A))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B2,aTP_Lamp_ld(set(A),fun(B,set(A)),A4)))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(product_prod(B,A),A)) ) ) ) ).

% card_of_Times_infinite_simps(3)
tff(fact_7998_regularCard__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca7133664381575040944arCard(A,R2)
    <=> ! [K5: set(A)] :
          ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),K5),aa(set(product_prod(A,A)),set(A),field2(A),R2))
            & bNF_Ca7293521722713021262ofinal(A,K5,R2) )
         => 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)),bNF_Ca6860139660246222851ard_of(A,K5)),R2),bNF_Wellorder_ordIso(A,A)) ) ) ).

% regularCard_def
tff(fact_7999_cardSuc__UNION__Cinfinite,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),As: fun(set(A),set(B)),B2: set(B)] :
      ( ( bNF_Ca4139267488887388095finite(A,R2)
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( bNF_Ca3754400796208372196lChain(set(A),set(B),bNF_Ca8387033319878233205ardSuc(A,R2),As)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(set(A)),set(set(B)),image2(set(A),set(B),As),aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))))
         => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),R2),bNF_Wellorder_ordLeq(B,A))
           => ? [X3: set(A)] :
                ( member(set(A),X3,aa(set(product_prod(set(A),set(A))),set(set(A)),field2(set(A)),bNF_Ca8387033319878233205ardSuc(A,R2)))
                & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B2),aa(set(A),set(B),As,X3)) ) ) ) ) ) ).

% cardSuc_UNION_Cinfinite
tff(fact_8000_cinfinite__def,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca4139267488887388095finite(A,R2)
    <=> ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2)) ) ).

% cinfinite_def
tff(fact_8001_Cinfinite__limit,axiom,
    ! [A: $tType,Xa: A,R2: set(product_prod(A,A))] :
      ( member(A,Xa,aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( ( bNF_Ca4139267488887388095finite(A,R2)
          & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
       => ? [X3: A] :
            ( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
            & ( Xa != X3 )
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa),X3),R2) ) ) ) ).

% Cinfinite_limit
tff(fact_8002_Cinfinite__limit2,axiom,
    ! [A: $tType,X15: A,R2: set(product_prod(A,A)),X23: A] :
      ( member(A,X15,aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( member(A,X23,aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( bNF_Ca4139267488887388095finite(A,R2)
            & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
         => ? [X3: A] :
              ( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
              & ( X15 != X3 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X15),X3),R2)
              & ( X23 != X3 )
              & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X23),X3),R2) ) ) ) ) ).

% Cinfinite_limit2
tff(fact_8003_card__of__Func__UNIV,axiom,
    ! [B: $tType,A: $tType,B2: set(B)] : member(product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),aa(set(product_prod(fun(A,B),fun(A,B))),product_prod(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B)))),product_Pair(set(product_prod(fun(A,B),fun(A,B))),set(product_prod(fun(A,B),fun(A,B))),bNF_Ca6860139660246222851ard_of(fun(A,B),bNF_Wellorder_Func(A,B,top_top(set(A)),B2))),bNF_Ca6860139660246222851ard_of(fun(A,B),aa(fun(fun(A,B),$o),set(fun(A,B)),collect(fun(A,B)),aTP_Lamp_akp(set(B),fun(fun(A,B),$o),B2)))),bNF_Wellorder_ordIso(fun(A,B),fun(A,B))) ).

% card_of_Func_UNIV
tff(fact_8004_card__of__bool,axiom,
    ! [A: $tType,A12: A,A23: A] :
      ( ( A12 != A23 )
     => member(product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod($o,$o)),set(product_prod(A,A))),product_Pair(set(product_prod($o,$o)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of($o,top_top(set($o)))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A12),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A23),bot_bot(set(A)))))),bNF_Wellorder_ordIso($o,A)) ) ).

% card_of_bool
tff(fact_8005_Un__Cinfinite__bound,axiom,
    ! [B: $tType,A: $tType,A4: set(A),R2: set(product_prod(B,B)),B2: set(A)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),R2),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,B2)),R2),bNF_Wellorder_ordLeq(A,B))
       => ( ( bNF_Ca4139267488887388095finite(B,R2)
            & bNF_Ca8970107618336181345der_on(B,aa(set(product_prod(B,B)),set(B),field2(B),R2),R2) )
         => member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))),R2),bNF_Wellorder_ordLeq(A,B)) ) ) ) ).

% Un_Cinfinite_bound
tff(fact_8006_card__of__Plus__empty2,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),aa(set(product_prod(sum_sum(B,A),sum_sum(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(B,A),sum_sum(B,A))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,bot_bot(set(B)),A4))),bNF_Wellorder_ordIso(A,sum_sum(B,A))) ).

% card_of_Plus_empty2
tff(fact_8007_card__of__Plus__empty1,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),aa(set(product_prod(sum_sum(A,B),sum_sum(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,B),sum_sum(A,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,bot_bot(set(B))))),bNF_Wellorder_ordIso(A,sum_sum(A,B))) ).

% card_of_Plus_empty1
tff(fact_8008_Cinfinite__limit__finite,axiom,
    ! [A: $tType,X4: set(A),R2: set(product_prod(A,A))] :
      ( aa(set(A),$o,finite_finite2(A),X4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X4),aa(set(product_prod(A,A)),set(A),field2(A),R2))
       => ( ( bNF_Ca4139267488887388095finite(A,R2)
            & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
         => ? [X3: A] :
              ( member(A,X3,aa(set(product_prod(A,A)),set(A),field2(A),R2))
              & ! [Xa2: A] :
                  ( member(A,Xa2,X4)
                 => ( ( Xa2 != X3 )
                    & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa2),X3),R2) ) ) ) ) ) ) ).

% Cinfinite_limit_finite
tff(fact_8009_card__of__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))
       => ( member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B2))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(sum_sum(A,B),A))
          & member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B2,A4))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ).

% card_of_Plus_infinite
tff(fact_8010_card__of__Plus__infinite1,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))
       => member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,A4,B2))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(sum_sum(A,B),A)) ) ) ).

% card_of_Plus_infinite1
tff(fact_8011_card__of__Plus__infinite2,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))
       => member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,B2,A4))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ).

% card_of_Plus_infinite2
tff(fact_8012_Card__order__Plus__infinite,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),P3: set(product_prod(B,B))] :
      ( ~ aa(set(A),$o,finite_finite2(A),aa(set(product_prod(A,A)),set(A),field2(A),R2))
     => ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),P3),R2),bNF_Wellorder_ordLeq(B,A))
         => ( member(product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(A,B),sum_sum(A,B))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(A,B),sum_Plus(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R2),aa(set(product_prod(B,B)),set(B),field2(B),P3)))),R2),bNF_Wellorder_ordIso(sum_sum(A,B),A))
            & member(product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A))),product_Pair(set(product_prod(sum_sum(B,A),sum_sum(B,A))),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(sum_sum(B,A),sum_Plus(B,A,aa(set(product_prod(B,B)),set(B),field2(B),P3),aa(set(product_prod(A,A)),set(A),field2(A),R2)))),R2),bNF_Wellorder_ordIso(sum_sum(B,A),A)) ) ) ) ) ).

% Card_order_Plus_infinite
tff(fact_8013_card__of__Times__infinite__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B2 != bot_bot(set(B)) )
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))
         => member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),aa(set(product_prod(product_prod(B,A),product_prod(B,A))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(B,A),product_prod(B,A))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(product_prod(B,A),product_Sigma(B,A,B2,aTP_Lamp_ld(set(A),fun(B,set(A)),A4)))),bNF_Wellorder_ordIso(A,product_prod(B,A))) ) ) ) ).

% card_of_Times_infinite_simps(4)
tff(fact_8014_card__of__Times__infinite__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,A4: set(A),B2: set(B)] :
      ( ~ aa(set(A),$o,finite_finite2(A),A4)
     => ( ( B2 != bot_bot(set(B)) )
       => ( member(product_prod(set(product_prod(B,B)),set(product_prod(A,A))),aa(set(product_prod(A,A)),product_prod(set(product_prod(B,B)),set(product_prod(A,A))),product_Pair(set(product_prod(B,B)),set(product_prod(A,A)),bNF_Ca6860139660246222851ard_of(B,B2)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Wellorder_ordLeq(B,A))
         => member(product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),aa(set(product_prod(product_prod(A,B),product_prod(A,B))),product_prod(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B)))),product_Pair(set(product_prod(A,A)),set(product_prod(product_prod(A,B),product_prod(A,B))),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Ca6860139660246222851ard_of(product_prod(A,B),product_Sigma(A,B,A4,aTP_Lamp_agb(set(B),fun(A,set(B)),B2)))),bNF_Wellorder_ordIso(A,product_prod(A,B))) ) ) ) ).

% card_of_Times_infinite_simps(2)
tff(fact_8015_Cnotzero__imp__not__empty,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( ( ~ 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)),R2),bNF_Cardinal_czero(A)),bNF_Wellorder_ordIso(A,A))
        & bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2) )
     => ( aa(set(product_prod(A,A)),set(A),field2(A),R2) != bot_bot(set(A)) ) ) ).

% Cnotzero_imp_not_empty
tff(fact_8016_czeroI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
      ( bNF_Ca8970107618336181345der_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( ( aa(set(product_prod(A,A)),set(A),field2(A),R2) = bot_bot(set(A)) )
       => member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B)) ) ) ).

% czeroI
tff(fact_8017_czero__def,axiom,
    ! [A: $tType] : bNF_Cardinal_czero(A) = bNF_Ca6860139660246222851ard_of(A,bot_bot(set(A))) ).

% czero_def
tff(fact_8018_czeroE,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),R2),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))
     => ( aa(set(product_prod(A,A)),set(A),field2(A),R2) = bot_bot(set(A)) ) ) ).

% czeroE
tff(fact_8019_card__of__ordIso__czero__iff__empty,axiom,
    ! [B: $tType,A: $tType,A4: set(A)] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),bNF_Ca6860139660246222851ard_of(A,A4)),bNF_Cardinal_czero(B)),bNF_Wellorder_ordIso(A,B))
    <=> ( A4 = bot_bot(set(A)) ) ) ).

% card_of_ordIso_czero_iff_empty
tff(fact_8020_cexp__mono2_H,axiom,
    ! [B: $tType,C: $tType,A: $tType,P22: set(product_prod(A,A)),R23: set(product_prod(B,B)),Q5: set(product_prod(C,C))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),P22),R23),bNF_Wellorder_ordLeq(A,B))
     => ( bNF_Ca8970107618336181345der_on(C,aa(set(product_prod(C,C)),set(C),field2(C),Q5),Q5)
       => ( ( ( aa(set(product_prod(A,A)),set(A),field2(A),P22) = bot_bot(set(A)) )
           => ( aa(set(product_prod(B,B)),set(B),field2(B),R23) = bot_bot(set(B)) ) )
         => member(product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),aa(set(product_prod(fun(B,C),fun(B,C))),product_prod(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C)))),product_Pair(set(product_prod(fun(A,C),fun(A,C))),set(product_prod(fun(B,C),fun(B,C))),bNF_Cardinal_cexp(C,A,Q5,P22)),bNF_Cardinal_cexp(C,B,Q5,R23)),bNF_Wellorder_ordLeq(fun(A,C),fun(B,C))) ) ) ) ).

% cexp_mono2'
tff(fact_8021_cexp__mono_H,axiom,
    ! [B: $tType,D: $tType,A: $tType,C: $tType,P12: set(product_prod(A,A)),R12: set(product_prod(B,B)),P22: set(product_prod(C,C)),R23: set(product_prod(D,D))] :
      ( member(product_prod(set(product_prod(A,A)),set(product_prod(B,B))),aa(set(product_prod(B,B)),product_prod(set(product_prod(A,A)),set(product_prod(B,B))),product_Pair(set(product_prod(A,A)),set(product_prod(B,B)),P12),R12),bNF_Wellorder_ordLeq(A,B))
     => ( member(product_prod(set(product_prod(C,C)),set(product_prod(D,D))),aa(set(product_prod(D,D)),product_prod(set(product_prod(C,C)),set(product_prod(D,D))),product_Pair(set(product_prod(C,C)),set(product_prod(D,D)),P22),R23),bNF_Wellorder_ordLeq(C,D))
       => ( ( ( aa(set(product_prod(C,C)),set(C),field2(C),P22) = bot_bot(set(C)) )
           => ( aa(set(product_prod(D,D)),set(D),field2(D),R23) = bot_bot(set(D)) ) )
         => member(product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),aa(set(product_prod(fun(D,B),fun(D,B))),product_prod(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B)))),product_Pair(set(product_prod(fun(C,A),fun(C,A))),set(product_prod(fun(D,B),fun(D,B))),bNF_Cardinal_cexp(A,C,P12,P22)),bNF_Cardinal_cexp(B,D,R12,R23)),bNF_Wellorder_ordLeq(fun(C,A),fun(D,B))) ) ) ) ).

% cexp_mono'
tff(fact_8022_Real_Opositive__def,axiom,
    positive2 = map_fun(real,fun(nat,rat),$o,$o,rep_real,id($o),aTP_Lamp_afp(fun(nat,rat),$o)) ).

% Real.positive_def
tff(fact_8023_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(complex,Z)),re(Z))),zero_zero(real))
    <=> ( re(Z) = aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Z)) ) ) ).

% cmod_plus_Re_le_0_iff
tff(fact_8024_id__funpow,axiom,
    ! [A: $tType,Na: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Na),id(A)) = id(A) ).

% id_funpow
tff(fact_8025_filtermap__id,axiom,
    ! [A: $tType] : aa(fun(A,A),fun(filter(A),filter(A)),filtermap(A,A),id(A)) = id(filter(A)) ).

% filtermap_id
tff(fact_8026_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_8027_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_8028_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_8029_complex__Re__le__cmod,axiom,
    ! [Xa: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(Xa)),real_V7770717601297561774m_norm(complex,Xa)) ).

% complex_Re_le_cmod
tff(fact_8030_abs__Re__le__cmod,axiom,
    ! [Xa: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),re(Xa))),real_V7770717601297561774m_norm(complex,Xa)) ).

% abs_Re_le_cmod
tff(fact_8031_Re__csqrt,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(csqrt(Z))) ).

% Re_csqrt
tff(fact_8032_ofilter__subset__embedS,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
          <=> bNF_Wellorder_embedS(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2))),id(A)) ) ) ) ) ).

% ofilter_subset_embedS
tff(fact_8033_Rat_Opositive__def,axiom,
    positive = map_fun(rat,product_prod(int,int),$o,$o,rep_Rat,id($o),aTP_Lamp_ahl(product_prod(int,int),$o)) ).

% Rat.positive_def
tff(fact_8034_rotate0,axiom,
    ! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).

% rotate0
tff(fact_8035_embedS__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),F2: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Wellorder_embedS(A,B,R2,R4,F2)
       => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(product_prod(A,A)),set(A),field2(A),R2))),aa(set(product_prod(B,B)),set(B),field2(B),R4)) ) ) ).

% embedS_Field
tff(fact_8036_ofilter__subset__embedS__iso,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B2)
         => ( ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),A4),B2)
            <=> bNF_Wellorder_embedS(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2))),id(A)) )
            & ( ( A4 = B2 )
            <=> bNF_Wellorder_iso(A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2))),id(A)) ) ) ) ) ) ).

% ofilter_subset_embedS_iso
tff(fact_8037_complex__abs__le__norm,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),real_V7770717601297561774m_norm(complex,Z))) ).

% complex_abs_le_norm
tff(fact_8038_csqrt__unique,axiom,
    ! [W2: complex,Z: complex] :
      ( ( aa(nat,complex,power_power(complex,W2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z )
     => ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(W2))
          | ( ( re(W2) = zero_zero(real) )
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(W2)) ) )
       => ( csqrt(Z) = W2 ) ) ) ).

% csqrt_unique
tff(fact_8039_csqrt__of__real__nonneg,axiom,
    ! [Xa: complex] :
      ( ( im(Xa) = zero_zero(real) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(Xa))
       => ( csqrt(Xa) = real_Vector_of_real(complex,aa(real,real,sqrt,re(Xa))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_8040_abs__Im__le__cmod,axiom,
    ! [Xa: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),im(Xa))),real_V7770717601297561774m_norm(complex,Xa)) ).

% abs_Im_le_cmod
tff(fact_8041_cmod__Im__le__iff,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( re(Xa) = re(Ya) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Xa)),real_V7770717601297561774m_norm(complex,Ya))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),im(Xa))),aa(real,real,abs_abs(real),im(Ya))) ) ) ).

% cmod_Im_le_iff
tff(fact_8042_cmod__Re__le__iff,axiom,
    ! [Xa: complex,Ya: complex] :
      ( ( im(Xa) = im(Ya) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Xa)),real_V7770717601297561774m_norm(complex,Ya))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),re(Xa))),aa(real,real,abs_abs(real),re(Ya))) ) ) ).

% cmod_Re_le_iff
tff(fact_8043_csqrt__principal,axiom,
    ! [Z: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(csqrt(Z)))
      | ( ( re(csqrt(Z)) = zero_zero(real) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(csqrt(Z))) ) ) ).

% csqrt_principal
tff(fact_8044_cmod__le,axiom,
    ! [Z: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z)))) ).

% cmod_le
tff(fact_8045_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% complex_neq_0
tff(fact_8046_csqrt__square,axiom,
    ! [B3: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(B3))
        | ( ( re(B3) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(B3)) ) )
     => ( csqrt(aa(nat,complex,power_power(complex,B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = B3 ) ) ).

% csqrt_square
tff(fact_8047_csqrt__of__real__nonpos,axiom,
    ! [Xa: complex] :
      ( ( im(Xa) = zero_zero(real) )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(Xa)),zero_zero(real))
       => ( csqrt(Xa) = 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(Xa))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_8048_csqrt__minus,axiom,
    ! [Xa: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(Xa)),zero_zero(real))
        | ( ( im(Xa) = zero_zero(real) )
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(Xa)) ) )
     => ( csqrt(aa(complex,complex,uminus_uminus(complex),Xa)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),csqrt(Xa)) ) ) ).

% csqrt_minus
tff(fact_8049_ofilter__subset__embed,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A),B2: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,A4)
       => ( order_ofilter(A,R2,B2)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
          <=> bNF_Wellorder_embed(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,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R2),product_Sigma(A,A,B2,aTP_Lamp_afz(set(A),fun(A,set(A)),B2))),id(A)) ) ) ) ) ).

% ofilter_subset_embed
tff(fact_8050_ofilter__embed,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),A4: set(A)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( order_ofilter(A,R2,A4)
      <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(product_prod(A,A)),set(A),field2(A),R2))
          & bNF_Wellorder_embed(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,A4,aTP_Lamp_afz(set(A),fun(A,set(A)),A4))),R2,id(A)) ) ) ) ).

% ofilter_embed
tff(fact_8051_embed__Field,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),F2: fun(A,B)] :
      ( bNF_Wellorder_embed(A,B,R2,R4,F2)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(product_prod(A,A)),set(A),field2(A),R2))),aa(set(product_prod(B,B)),set(B),field2(B),R4)) ) ).

% embed_Field
tff(fact_8052_embedS__iff,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),F2: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Wellorder_embed(A,B,R2,R4,F2)
       => ( bNF_Wellorder_embedS(A,B,R2,R4,F2)
        <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less(set(B)),aa(set(A),set(B),image2(A,B,F2),aa(set(product_prod(A,A)),set(A),field2(A),R2))),aa(set(product_prod(B,B)),set(B),field2(B),R4)) ) ) ) ).

% embedS_iff
tff(fact_8053_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N4: nat,F2: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N2: nat] : member(complex,aa(nat,complex,G,N2),real_Vector_Reals(complex))
           => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G,N2)))
             => ( ! [N2: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N4),N2)
                   => aa(real,$o,aa(real,fun(real,$o),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_8054_Rangep__Range__eq,axiom,
    ! [A: $tType,B: $tType,R2: set(product_prod(B,A)),X2: A] :
      ( aa(A,$o,rangep(B,A,aTP_Lamp_aci(set(product_prod(B,A)),fun(B,fun(A,$o)),R2)),X2)
    <=> member(A,X2,aa(set(product_prod(B,A)),set(A),range(B,A),R2)) ) ).

% Rangep_Range_eq
tff(fact_8055_nonzero__Reals__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: A] :
          ( member(A,A3,real_Vector_Reals(A))
         => ( ( A3 != zero_zero(A) )
           => member(A,aa(A,A,inverse_inverse(A),A3),real_Vector_Reals(A)) ) ) ) ).

% nonzero_Reals_inverse
tff(fact_8056_Reals__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => member(A,zero_zero(A),real_Vector_Reals(A)) ) ).

% Reals_0
tff(fact_8057_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A3: A,B3: A] :
          ( member(A,A3,real_Vector_Reals(A))
         => ( member(A,B3,real_Vector_Reals(A))
           => ( ( B3 != zero_zero(A) )
             => member(A,divide_divide(A,A3,B3),real_Vector_Reals(A)) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_8058_Rangep_Ocases,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: B] :
      ( aa(B,$o,rangep(A,B,R2),A3)
     => ~ ! [A5: A] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A5),A3) ) ).

% Rangep.cases
tff(fact_8059_Rangep_Osimps,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: B] :
      ( aa(B,$o,rangep(A,B,R2),A3)
    <=> ? [A9: A,B7: B] :
          ( ( A3 = B7 )
          & aa(B,$o,aa(A,fun(B,$o),R2,A9),B7) ) ) ).

% Rangep.simps
tff(fact_8060_RangePI,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),R2,A3),B3)
     => aa(B,$o,rangep(A,B,R2),B3) ) ).

% RangePI
tff(fact_8061_RangepE,axiom,
    ! [A: $tType,B: $tType,R2: fun(A,fun(B,$o)),B3: B] :
      ( aa(B,$o,rangep(A,B,R2),B3)
     => ~ ! [A5: A] : ~ aa(B,$o,aa(A,fun(B,$o),R2,A5),B3) ) ).

% RangepE
tff(fact_8062_Range__def,axiom,
    ! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(B),range(A,B),X2) = aa(fun(B,$o),set(B),collect(B),rangep(A,B,aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),X2))) ).

% Range_def
tff(fact_8063_num_Orec__transfer,axiom,
    ! [A: $tType,B: $tType,S: fun(A,fun(B,$o))] : aa(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),$o,aa(fun(A,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A)))),fun(fun(B,fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B)))),$o),bNF_rel_fun(A,B,fun(fun(num,fun(A,A)),fun(fun(num,fun(A,A)),fun(num,A))),fun(fun(num,fun(B,B)),fun(fun(num,fun(B,B)),fun(num,B))),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_8064_complex__div__gt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(divide_divide(complex,A3,B3)))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) )
      & ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(divide_divide(complex,A3,B3)))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ) ).

% complex_div_gt_0
tff(fact_8065_Re__complex__div__gt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(divide_divide(complex,A3,B3)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ).

% Re_complex_div_gt_0
tff(fact_8066_Re__complex__div__lt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),re(divide_divide(complex,A3,B3))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))),zero_zero(real)) ) ).

% Re_complex_div_lt_0
tff(fact_8067_Re__complex__div__le__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(divide_divide(complex,A3,B3))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))),zero_zero(real)) ) ).

% Re_complex_div_le_0
tff(fact_8068_Re__complex__div__ge__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(divide_divide(complex,A3,B3)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ).

% Re_complex_div_ge_0
tff(fact_8069_Im__complex__div__gt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(divide_divide(complex,A3,B3)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ).

% Im_complex_div_gt_0
tff(fact_8070_Im__complex__div__lt__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(divide_divide(complex,A3,B3))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))),zero_zero(real)) ) ).

% Im_complex_div_lt_0
tff(fact_8071_Im__complex__div__le__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(divide_divide(complex,A3,B3))),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))),zero_zero(real)) ) ).

% Im_complex_div_le_0
tff(fact_8072_Im__complex__div__ge__0,axiom,
    ! [A3: complex,B3: complex] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(divide_divide(complex,A3,B3)))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A3),cnj(B3)))) ) ).

% Im_complex_div_ge_0
tff(fact_8073_num_Ocase__transfer,axiom,
    ! [A: $tType,B: $tType,S: fun(A,fun(B,$o))] : aa(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),$o,aa(fun(A,fun(fun(num,A),fun(fun(num,A),fun(num,A)))),fun(fun(B,fun(fun(num,B),fun(fun(num,B),fun(num,B)))),$o),bNF_rel_fun(A,B,fun(fun(num,A),fun(fun(num,A),fun(num,A))),fun(fun(num,B),fun(fun(num,B),fun(num,B))),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_8074_Zfun__imp__Zfun,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F2: fun(A,B),F3: filter(A),G: fun(A,C),K4: real] :
          ( zfun(A,B,F2,F3)
         => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_uj(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),F2),G),K4)),F3)
           => zfun(A,C,G,F3) ) ) ) ).

% Zfun_imp_Zfun
tff(fact_8075_Zfun__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: filter(A)] : zfun(A,B,aTP_Lamp_akq(A,B),F3) ) ).

% Zfun_zero
tff(fact_8076_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F13: B,F24: fun(num,B),F33: fun(num,B),Num: num] : aa(B,A,H,aa(num,B,aa(fun(num,B),fun(num,B),aa(fun(num,B),fun(fun(num,B),fun(num,B)),aa(B,fun(fun(num,B),fun(fun(num,B),fun(num,B))),case_num(B),F13),F24),F33),Num)) = aa(num,A,aa(fun(num,A),fun(num,A),aa(fun(num,A),fun(fun(num,A),fun(num,A)),aa(A,fun(fun(num,A),fun(fun(num,A),fun(num,A))),case_num(A),aa(B,A,H,F13)),aa(fun(num,B),fun(num,A),aTP_Lamp_akr(fun(B,A),fun(fun(num,B),fun(num,A)),H),F24)),aa(fun(num,B),fun(num,A),aTP_Lamp_akr(fun(B,A),fun(fun(num,B),fun(num,A)),H),F33)),Num) ).

% num.case_distrib
tff(fact_8077_Zfun__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,B),F3: filter(A),F2: fun(A,C)] :
          ( zfun(A,B,G,F3)
         => ( ! [X3: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,F2,X3))),real_V7770717601297561774m_norm(B,aa(A,B,G,X3)))
           => zfun(A,C,F2,F3) ) ) ) ).

% Zfun_le
tff(fact_8078_Zfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( zfun(A,B,F2,F3)
        <=> ! [R5: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R5)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(real,fun(A,$o)),F2),R5)),F3) ) ) ) ).

% Zfun_def
tff(fact_8079_ZfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A)] :
          ( ! [R3: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R3)
             => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(real,fun(A,$o)),F2),R3)),F3) )
         => zfun(A,B,F2,F3) ) ) ).

% ZfunI
tff(fact_8080_ZfunD,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F2: fun(A,B),F3: filter(A),R2: real] :
          ( zfun(A,B,F2,F3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),R2)
           => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(real,fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(real,fun(A,$o)),F2),R2)),F3) ) ) ) ).

% ZfunD
tff(fact_8081_semilattice__order__set_Osubset__imp,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),B2: set(A)] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),B2)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(set(A),A,lattic1715443433743089157tice_F(A,F2),B2)),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4)) ) ) ) ) ).

% semilattice_order_set.subset_imp
tff(fact_8082_MOST__INFM,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(set(A),$o,finite_finite2(A),top_top(set(A)))
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A)) ) ) ).

% MOST_INFM
tff(fact_8083_not__MOST,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)),cofinite(A)) ) ).

% not_MOST
tff(fact_8084_not__INFM,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
    <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)),cofinite(A)) ) ).

% not_INFM
tff(fact_8085_cofinite__bot,axiom,
    ! [A: $tType] :
      ( ( cofinite(A) = bot_bot(filter(A)) )
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% cofinite_bot
tff(fact_8086_MOST__const,axiom,
    ! [A: $tType,P: $o] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_mb($o,fun(A,$o),(P))),cofinite(A))
    <=> ( (P)
        | aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% MOST_const
tff(fact_8087_MOST__eq_I1_J,axiom,
    ! [A: $tType,A3: A] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aTP_Lamp_ae(A,fun(A,$o),A3)),cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% MOST_eq(1)
tff(fact_8088_MOST__eq_I2_J,axiom,
    ! [A: $tType,A3: A] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),fequal(A),A3)),cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% MOST_eq(2)
tff(fact_8089_INFM__neq_I2_J,axiom,
    ! [A: $tType,A3: A] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(A,fun(A,$o),aTP_Lamp_ahw(A,fun(A,$o)),A3)),cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% INFM_neq(2)
tff(fact_8090_INFM__neq_I1_J,axiom,
    ! [A: $tType,A3: A] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(A,fun(A,$o),aTP_Lamp_adi(A,fun(A,$o)),A3)),cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ).

% INFM_neq(1)
tff(fact_8091_INFM__const,axiom,
    ! [A: $tType,P: $o] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_mb($o,fun(A,$o),(P))),cofinite(A))
    <=> ( (P)
        & ~ aa(set(A),$o,finite_finite2(A),top_top(set(A))) ) ) ).

% INFM_const
tff(fact_8092_MOST__ge__nat,axiom,
    ! [M: nat] : aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aa(nat,fun(nat,$o),ord_less_eq(nat),M)),cofinite(nat)) ).

% MOST_ge_nat
tff(fact_8093_MOST__nat__le,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),cofinite(nat))
    <=> ? [M2: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
         => aa(nat,$o,P,N) ) ) ).

% MOST_nat_le
tff(fact_8094_cofinite__def,axiom,
    ! [A: $tType] : cofinite(A) = abs_filter(A,aTP_Lamp_akt(fun(A,$o),$o)) ).

% cofinite_def
tff(fact_8095_semilattice__order__set_OcoboundedI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),A3: A] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( member(A,A3,A4)
         => aa(A,$o,aa(A,fun(A,$o),Less_eq,aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4)),A3) ) ) ) ).

% semilattice_order_set.coboundedI
tff(fact_8096_Sup__fin__def,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ( lattic5882676163264333800up_fin(A) = lattic1715443433743089157tice_F(A,sup_sup(A)) ) ) ).

% Sup_fin_def
tff(fact_8097_eventually__cofinite,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P))) ) ).

% eventually_cofinite
tff(fact_8098_MOST__iff__finiteNeg,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P))) ) ).

% MOST_iff_finiteNeg
tff(fact_8099_MOST__Suc__iff,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_ic(fun(nat,$o),fun(nat,$o),P)),cofinite(nat))
    <=> aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),cofinite(nat)) ) ).

% MOST_Suc_iff
tff(fact_8100_MOST__SucI,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),cofinite(nat))
     => aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_ic(fun(nat,$o),fun(nat,$o),P)),cofinite(nat)) ) ).

% MOST_SucI
tff(fact_8101_MOST__SucD,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),aTP_Lamp_ic(fun(nat,$o),fun(nat,$o),P)),cofinite(nat))
     => aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),cofinite(nat)) ) ).

% MOST_SucD
tff(fact_8102_Max__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattic643756798349783984er_Max(A) = lattic1715443433743089157tice_F(A,ord_max(A)) ) ) ).

% Max_def
tff(fact_8103_Min__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattic643756798350308766er_Min(A) = lattic1715443433743089157tice_F(A,ord_min(A)) ) ) ).

% Min_def
tff(fact_8104_cofinite__eq__sequentially,axiom,
    cofinite(nat) = at_top(nat) ).

% cofinite_eq_sequentially
tff(fact_8105_semilattice__set_OF_Ocong,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A))] : lattic1715443433743089157tice_F(A,F2) = lattic1715443433743089157tice_F(A,F2) ).

% semilattice_set.F.cong
tff(fact_8106_MOST__neq_I2_J,axiom,
    ! [A: $tType,A3: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),aTP_Lamp_ahw(A,fun(A,$o)),A3)),cofinite(A)) ).

% MOST_neq(2)
tff(fact_8107_MOST__neq_I1_J,axiom,
    ! [A: $tType,A3: A] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(A,fun(A,$o),aTP_Lamp_adi(A,fun(A,$o)),A3)),cofinite(A)) ).

% MOST_neq(1)
tff(fact_8108_MOST__eq__imp_I2_J,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aku(A,fun(fun(A,$o),fun(A,$o)),A3),P)),cofinite(A)) ).

% MOST_eq_imp(2)
tff(fact_8109_MOST__eq__imp_I1_J,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] : aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_akv(A,fun(fun(A,$o),fun(A,$o)),A3),P)),cofinite(A)) ).

% MOST_eq_imp(1)
tff(fact_8110_MOST__I,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X3: A] : aa(A,$o,P,X3)
     => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A)) ) ).

% MOST_I
tff(fact_8111_ALL__MOST,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_1: A] : aa(A,$o,P,X_1)
     => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A)) ) ).

% ALL_MOST
tff(fact_8112_MOST__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
     => ( ! [X3: A] :
            ( aa(A,$o,P,X3)
           => aa(A,$o,Q,X3) )
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),cofinite(A)) ) ) ).

% MOST_mono
tff(fact_8113_MOST__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),cofinite(A))
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),cofinite(A)) ) ) ).

% MOST_conjI
tff(fact_8114_MOST__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),cofinite(A))
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),cofinite(A)) ) ) ).

% MOST_rev_mp
tff(fact_8115_MOST__imp__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),cofinite(A))
      <=> aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),cofinite(A)) ) ) ).

% MOST_imp_iff
tff(fact_8116_MOST__conj__distrib,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),cofinite(A))
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),cofinite(A)) ) ) ).

% MOST_conj_distrib
tff(fact_8117_Inf__fin__def,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ( lattic7752659483105999362nf_fin(A) = lattic1715443433743089157tice_F(A,inf_inf(A)) ) ) ).

% Inf_fin_def
tff(fact_8118_MOST__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),eventually(nat),P),cofinite(nat))
    <=> ? [M2: nat] :
        ! [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
         => aa(nat,$o,P,N) ) ) ).

% MOST_nat
tff(fact_8119_INFM__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),frequently(nat),P),cofinite(nat))
    <=> ! [M2: nat] :
        ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),M2),N)
          & aa(nat,$o,P,N) ) ) ).

% INFM_nat
tff(fact_8120_INFM__imp__distrib,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),cofinite(A))
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),cofinite(A)) ) ) ).

% INFM_imp_distrib
tff(fact_8121_Alm__all__def,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
    <=> ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),P)),cofinite(A)) ) ).

% Alm_all_def
tff(fact_8122_INFM__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
     => ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q),cofinite(A))
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),cofinite(A)) ) ) ).

% INFM_conjI
tff(fact_8123_not__INFM__eq_I2_J,axiom,
    ! [A: $tType,A3: A] : ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(A,fun(A,$o),fequal(A),A3)),cofinite(A)) ).

% not_INFM_eq(2)
tff(fact_8124_not__INFM__eq_I1_J,axiom,
    ! [A: $tType,A3: A] : ~ aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aTP_Lamp_ae(A,fun(A,$o),A3)),cofinite(A)) ).

% not_INFM_eq(1)
tff(fact_8125_INFM__E,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
     => ~ ! [X3: A] : ~ aa(A,$o,P,X3) ) ).

% INFM_E
tff(fact_8126_INFM__EX,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% INFM_EX
tff(fact_8127_INFM__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
     => ( ! [X3: A] :
            ( aa(A,$o,P,X3)
           => aa(A,$o,Q,X3) )
       => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),cofinite(A)) ) ) ).

% INFM_mono
tff(fact_8128_INFM__disj__distrib,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)),cofinite(A))
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
        | aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),Q),cofinite(A)) ) ) ).

% INFM_disj_distrib
tff(fact_8129_frequently__cofinite,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P)) ) ).

% frequently_cofinite
tff(fact_8130_INFM__iff__infinite,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),P),cofinite(A))
    <=> ~ aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),P)) ) ).

% INFM_iff_infinite
tff(fact_8131_INFM__nat__le,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(filter(nat),$o,aa(fun(nat,$o),fun(filter(nat),$o),frequently(nat),P),cofinite(nat))
    <=> ! [M2: nat] :
        ? [N: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N)
          & aa(nat,$o,P,N) ) ) ).

% INFM_nat_le
tff(fact_8132_MOST__finite__Ball__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(A,fun(B,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,fun(B,$o)),fun(B,$o),aTP_Lamp_akw(set(A),fun(fun(A,fun(B,$o)),fun(B,$o)),A4),P)),cofinite(B))
      <=> ! [X: A] :
            ( member(A,X,A4)
           => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(A,fun(B,$o),P,X)),cofinite(B)) ) ) ) ).

% MOST_finite_Ball_distrib
tff(fact_8133_MOST__inj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F2: fun(B,A)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),P),cofinite(A))
     => ( inj_on(B,A,F2,top_top(set(B)))
       => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(B,A),fun(B,$o),aTP_Lamp_sr(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),F2)),cofinite(B)) ) ) ).

% MOST_inj
tff(fact_8134_semilattice__order__set_OboundedE,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),Xa: A] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4))
           => ! [A10: A] :
                ( member(A,A10,A4)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),A10) ) ) ) ) ) ).

% semilattice_order_set.boundedE
tff(fact_8135_semilattice__order__set_OboundedI,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),Xa: A] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( ! [A5: A] :
                ( member(A,A5,A4)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),A5) )
           => aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4)) ) ) ) ) ).

% semilattice_order_set.boundedI
tff(fact_8136_semilattice__order__set_Obounded__iff,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o)),A4: set(A),Xa: A] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4))
          <=> ! [X: A] :
                ( member(A,X,A4)
               => aa(A,$o,aa(A,fun(A,$o),Less_eq,Xa),X) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
tff(fact_8137_INFM__inj,axiom,
    ! [A: $tType,B: $tType,P: fun(B,$o),F2: fun(A,B)] :
      ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),frequently(A),aa(fun(A,B),fun(A,$o),aTP_Lamp_akx(fun(B,$o),fun(fun(A,B),fun(A,$o)),P),F2)),cofinite(A))
     => ( inj_on(A,B,F2,top_top(set(A)))
       => aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),P),cofinite(B)) ) ) ).

% INFM_inj
tff(fact_8138_INFM__finite__Bex__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(A,fun(B,$o))] :
      ( aa(set(A),$o,finite_finite2(A),A4)
     => ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(fun(A,fun(B,$o)),fun(B,$o),aTP_Lamp_aky(set(A),fun(fun(A,fun(B,$o)),fun(B,$o)),A4),P)),cofinite(B))
      <=> ? [X: A] :
            ( member(A,X,A4)
            & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),frequently(B),aa(A,fun(B,$o),P,X)),cofinite(B)) ) ) ) ).

% INFM_finite_Bex_distrib
tff(fact_8139_semilattice__set_Oeq__fold_H,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4) = the2(A,finite_fold(A,option(A),aTP_Lamp_akz(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),F2),none(A),A4)) ) ) ).

% semilattice_set.eq_fold'
tff(fact_8140_semilattice__set_Oinsert__remove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),F2,Xa),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ).

% semilattice_set.insert_remove
tff(fact_8141_semilattice__set_Oin__idem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( member(A,Xa,A4)
         => ( aa(A,A,aa(A,fun(A,A),F2,Xa),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4) ) ) ) ) ).

% semilattice_set.in_idem
tff(fact_8142_semilattice__order__set_Oaxioms_I2_J,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Less_eq: fun(A,fun(A,$o)),Less: fun(A,fun(A,$o))] :
      ( lattic4895041142388067077er_set(A,F2,Less_eq,Less)
     => lattic149705377957585745ce_set(A,F2) ) ).

% semilattice_order_set.axioms(2)
tff(fact_8143_Inf__fin_Osemilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => lattic149705377957585745ce_set(A,inf_inf(A)) ) ).

% Inf_fin.semilattice_set_axioms
tff(fact_8144_Max_Osemilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic149705377957585745ce_set(A,ord_max(A)) ) ).

% Max.semilattice_set_axioms
tff(fact_8145_Min_Osemilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => lattic149705377957585745ce_set(A,ord_min(A)) ) ).

% Min.semilattice_set_axioms
tff(fact_8146_Sup__fin_Osemilattice__set__axioms,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => lattic149705377957585745ce_set(A,sup_sup(A)) ) ).

% Sup_fin.semilattice_set_axioms
tff(fact_8147_semilattice__set_Osingleton,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = Xa ) ) ).

% semilattice_set.singleton
tff(fact_8148_semilattice__set_Ohom__commute,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),H: fun(A,A),N4: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( ! [X3: A,Y: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),F2,X3),Y)) = aa(A,A,aa(A,fun(A,A),F2,aa(A,A,H,X3)),aa(A,A,H,Y))
       => ( aa(set(A),$o,finite_finite2(A),N4)
         => ( ( N4 != bot_bot(set(A)) )
           => ( aa(A,A,H,aa(set(A),A,lattic1715443433743089157tice_F(A,F2),N4)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),image2(A,A,H),N4)) ) ) ) ) ) ).

% semilattice_set.hom_commute
tff(fact_8149_semilattice__set_Osubset,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),B2: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( B2 != bot_bot(set(A)) )
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
           => ( aa(A,A,aa(A,fun(A,A),F2,aa(set(A),A,lattic1715443433743089157tice_F(A,F2),B2)),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4)) = aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4) ) ) ) ) ) ).

% semilattice_set.subset
tff(fact_8150_semilattice__set_Oinsert__not__elem,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ~ member(A,Xa,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),F2,Xa),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4)) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
tff(fact_8151_semilattice__set_Oinsert,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = aa(A,A,aa(A,fun(A,A),F2,Xa),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4)) ) ) ) ) ).

% semilattice_set.insert
tff(fact_8152_semilattice__set_Oclosed,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( ! [X3: A,Y: A] : member(A,aa(A,A,aa(A,fun(A,A),F2,X3),Y),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Y),bot_bot(set(A)))))
           => member(A,aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4),A4) ) ) ) ) ).

% semilattice_set.closed
tff(fact_8153_semilattice__set_Ounion,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),B2: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( ( A4 != bot_bot(set(A)) )
         => ( aa(set(A),$o,finite_finite2(A),B2)
           => ( ( B2 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2)) = aa(A,A,aa(A,fun(A,A),F2,aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4)),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),B2)) ) ) ) ) ) ) ).

% semilattice_set.union
tff(fact_8154_semilattice__set_Oeq__fold,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),A4)) = finite_fold(A,A,F2,Xa,A4) ) ) ) ).

% semilattice_set.eq_fold
tff(fact_8155_semilattice__set_Oinfinite,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A)] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( ~ aa(set(A),$o,finite_finite2(A),A4)
       => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4) = the2(A,none(A)) ) ) ) ).

% semilattice_set.infinite
tff(fact_8156_semilattice__set_Oremove,axiom,
    ! [A: $tType,F2: fun(A,fun(A,A)),A4: set(A),Xa: A] :
      ( lattic149705377957585745ce_set(A,F2)
     => ( aa(set(A),$o,finite_finite2(A),A4)
       => ( member(A,Xa,A4)
         => ( aa(set(A),A,lattic1715443433743089157tice_F(A,F2),A4) = $ite(aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A)))) = bot_bot(set(A)),Xa,aa(A,A,aa(A,fun(A,A),F2,Xa),aa(set(A),A,lattic1715443433743089157tice_F(A,F2),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Xa),bot_bot(set(A))))))) ) ) ) ) ).

% semilattice_set.remove
tff(fact_8157_folding__def_H,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite_folding(A,B,F2)
    <=> finite_folding_on(A,B,top_top(set(A)),F2) ) ).

% folding_def'
tff(fact_8158_antisymp__antisym__eq,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] :
      ( antisymp(A,aTP_Lamp_ahx(set(product_prod(A,A)),fun(A,fun(A,$o)),R2))
    <=> antisym(A,R2) ) ).

% antisymp_antisym_eq
tff(fact_8159_antisym__bot,axiom,
    ! [A: $tType] : antisymp(A,bot_bot(fun(A,fun(A,$o)))) ).

% antisym_bot
tff(fact_8160_antisymp__less__eq,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),S3: fun(A,fun(A,$o))] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R2),S3)
     => ( antisymp(A,S3)
       => antisymp(A,R2) ) ) ).

% antisymp_less_eq
tff(fact_8161_folding__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite_folding(A,B,F2)
    <=> ! [Y4: A,X: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y4)),aa(A,fun(B,B),F2,X)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,Y4)) ) ).

% folding_def
tff(fact_8162_folding_Ocomp__fun__commute,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B)),Ya: A,Xa: A] :
      ( finite_folding(A,B,F2)
     => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Ya)),aa(A,fun(B,B),F2,Xa)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Xa)),aa(A,fun(B,B),F2,Ya)) ) ) ).

% folding.comp_fun_commute
tff(fact_8163_folding_Ointro,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( ! [Y: A,X3: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,Y)),aa(A,fun(B,B),F2,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,Y))
     => finite_folding(A,B,F2) ) ).

% folding.intro
tff(fact_8164_card_Ofolding__axioms,axiom,
    ! [A: $tType] : finite_folding(A,nat,aTP_Lamp_mh(A,fun(nat,nat))) ).

% card.folding_axioms
tff(fact_8165_antisymp__equality,axiom,
    ! [A: $tType] : antisymp(A,fequal(A)) ).

% antisymp_equality
tff(fact_8166_antisymp__def,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( antisymp(A,R2)
    <=> ! [X: A,Y4: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,X),Y4)
         => ( aa(A,$o,aa(A,fun(A,$o),R2,Y4),X)
           => ( X = Y4 ) ) ) ) ).

% antisymp_def
tff(fact_8167_antisympI,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o))] :
      ( ! [X3: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),R2,X3),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),R2,Y),X3)
           => ( X3 = Y ) ) )
     => antisymp(A,R2) ) ).

% antisympI
tff(fact_8168_antisympD,axiom,
    ! [A: $tType,R2: fun(A,fun(A,$o)),A3: A,B3: A] :
      ( antisymp(A,R2)
     => ( aa(A,$o,aa(A,fun(A,$o),R2,A3),B3)
       => ( aa(A,$o,aa(A,fun(A,$o),R2,B3),A3)
         => ( A3 = B3 ) ) ) ) ).

% antisympD
tff(fact_8169_folding__idem_Oaxioms_I1_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite_folding_idem(A,B,F2)
     => finite_folding(A,B,F2) ) ).

% folding_idem.axioms(1)
tff(fact_8170_folding__idem_Ointro,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite_folding(A,B,F2)
     => ( finite7837460588564673216axioms(A,B,F2)
       => finite_folding_idem(A,B,F2) ) ) ).

% folding_idem.intro
tff(fact_8171_folding__idem__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite_folding_idem(A,B,F2)
    <=> ( finite_folding(A,B,F2)
        & finite7837460588564673216axioms(A,B,F2) ) ) ).

% folding_idem_def
tff(fact_8172_folding__idem__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( ! [X3: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X3)),aa(A,fun(B,B),F2,X3)) = aa(A,fun(B,B),F2,X3)
     => finite7837460588564673216axioms(A,B,F2) ) ).

% folding_idem_axioms.intro
tff(fact_8173_folding__idem__axioms__def,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite7837460588564673216axioms(A,B,F2)
    <=> ! [X: A] : aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F2,X)),aa(A,fun(B,B),F2,X)) = aa(A,fun(B,B),F2,X) ) ).

% folding_idem_axioms_def
tff(fact_8174_folding__idem_Oaxioms_I2_J,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,fun(B,B))] :
      ( finite_folding_idem(A,B,F2)
     => finite7837460588564673216axioms(A,B,F2) ) ).

% folding_idem.axioms(2)
tff(fact_8175_inv__into__Field__embed,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,A)),R4: set(product_prod(B,B)),F2: fun(A,B)] :
      ( order_well_order_on(A,aa(set(product_prod(A,A)),set(A),field2(A),R2),R2)
     => ( bNF_Wellorder_embed(A,B,R2,R4,F2)
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(product_prod(B,B)),set(B),field2(B),R4)),aa(set(A),set(B),image2(A,B,F2),aa(set(product_prod(A,A)),set(A),field2(A),R2)))
         => bNF_Wellorder_embed(B,A,R4,R2,hilbert_inv_into(A,B,aa(set(product_prod(A,A)),set(A),field2(A),R2),F2)) ) ) ) ).

% inv_into_Field_embed
tff(fact_8176_Un__csum,axiom,
    ! [A: $tType,A4: set(A),B2: set(A)] : member(product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),aa(set(product_prod(sum_sum(A,A),sum_sum(A,A))),product_prod(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A)))),product_Pair(set(product_prod(A,A)),set(product_prod(sum_sum(A,A),sum_sum(A,A))),bNF_Ca6860139660246222851ard_of(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A4),B2))),bNF_Cardinal_csum(A,A,bNF_Ca6860139660246222851ard_of(A,A4),bNF_Ca6860139660246222851ard_of(A,B2))),bNF_Wellorder_ordLeq(A,sum_sum(A,A))) ).

% Un_csum
tff(fact_8177_inv__into__image__cancel,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),S: set(A)] :
      ( inj_on(A,B,F2,A4)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S),A4)
       => ( aa(set(B),set(A),image2(B,A,hilbert_inv_into(A,B,A4,F2)),aa(set(A),set(B),image2(A,B,F2),S)) = S ) ) ) ).

% inv_into_image_cancel
tff(fact_8178_image__inv__into__cancel,axiom,
    ! [B: $tType,A: $tType,F2: fun(B,A),A4: set(B),A14: set(A),B13: set(A)] :
      ( ( aa(set(B),set(A),image2(B,A,F2),A4) = A14 )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B13),A14)
       => ( aa(set(B),set(A),image2(B,A,F2),aa(set(A),set(B),image2(A,B,hilbert_inv_into(B,A,A4,F2)),B13)) = B13 ) ) ) ).

% image_inv_into_cancel
tff(fact_8179_bij__betw__inv__into__subset,axiom,
    ! [B: $tType,A: $tType,F2: fun(A,B),A4: set(A),A14: set(B),B2: set(A),B13: set(B)] :
      ( bij_betw(A,B,F2,A4,A14)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),A4)
       => ( ( aa(set(A),set(B),image2(A,B,F2),B2) = B13 )
         => bij_betw(B,A,hilbert_inv_into(A,B,A4,F2),B13,B2) ) ) ) ).

% bij_betw_inv_into_subset
tff(fact_8180_inj__on__inv__into,axiom,
    ! [B: $tType,A: $tType,B2: set(A),F2: fun(B,A),A4: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B2),aa(set(B),set(A),image2(B,A,F2),A4))
     => inj_on(A,B,hilbert_inv_into(B,A,A4,F2),B2) ) ).

% inj_on_inv_into
tff(fact_8181_strict__sorted__equal__Uniq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : uniq(list(A),aTP_Lamp_ala(set(A),fun(list(A),$o),A4)) ) ).

% strict_sorted_equal_Uniq
tff(fact_8182_subset__singleton__iff__Uniq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( ? [A9: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A9),bot_bot(set(A))))
    <=> uniq(A,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),A4)) ) ).

% subset_singleton_iff_Uniq
tff(fact_8183_single__valuedp__iff__Uniq,axiom,
    ! [B: $tType,A: $tType,R2: fun(A,fun(B,$o))] :
      ( single_valuedp(A,B,R2)
    <=> ! [X: A] : uniq(B,aa(A,fun(B,$o),R2,X)) ) ).

% single_valuedp_iff_Uniq
tff(fact_8184_ATP_Olambda__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_qj(A,A),Uu) = divide_divide(A,aa(A,A,minus_minus(A,exp(A,Uu)),one_one(A)),Uu) ) ).

% ATP.lambda_1
tff(fact_8185_ATP_Olambda__2,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_iy(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(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_2
tff(fact_8186_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(product_prod(A,A)),aTP_Lamp_mt(A,set(product_prod(A,A))),Uu) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Uu),Uu)),bot_bot(set(product_prod(A,A)))) ).

% ATP.lambda_3
tff(fact_8187_ATP_Olambda__4,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_abh(A,$o),Uu)
        <=> ( member(A,Uu,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Uu) ) ) ) ).

% ATP.lambda_4
tff(fact_8188_ATP_Olambda__5,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_gl(real,$o),Uu)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_5
tff(fact_8189_ATP_Olambda__6,axiom,
    ! [A: $tType,Uu: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_adz(set(A),$o),Uu)
    <=> ( ( Uu != bot_bot(set(A)) )
        & countable_countable(A,Uu) ) ) ).

% ATP.lambda_6
tff(fact_8190_ATP_Olambda__7,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_ahf(product_prod(A,A),$o),Uu)
    <=> ( aa(product_prod(A,A),A,product_fst(A,A),Uu) = aa(product_prod(A,A),A,product_snd(A,A),Uu) ) ) ).

% ATP.lambda_7
tff(fact_8191_ATP_Olambda__8,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] :
          ( aa(nat,$o,aTP_Lamp_jd(nat,$o),Uu)
        <=> ( aa(nat,A,semiring_1_of_nat(A),Uu) = zero_zero(A) ) ) ) ).

% ATP.lambda_8
tff(fact_8192_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_xw(nat,nat),Uu) = aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_9
tff(fact_8193_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_gf(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_10
tff(fact_8194_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,product_prod(nat,nat),aTP_Lamp_afq(nat,product_prod(nat,nat)),Uu) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Uu),zero_zero(nat)) ).

% ATP.lambda_11
tff(fact_8195_ATP_Olambda__12,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_lf(A,set(A)),Uu) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))) ).

% ATP.lambda_12
tff(fact_8196_ATP_Olambda__13,axiom,
    ! [Uu: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_ahl(product_prod(int,int),$o),Uu)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),Uu)),aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_13
tff(fact_8197_ATP_Olambda__14,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_rb(nat,A),Uu) = divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_14
tff(fact_8198_ATP_Olambda__15,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ado(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,minus_minus(nat,aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_15
tff(fact_8199_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(set(product_prod(A,A)),set(product_prod(A,A))),aTP_Lamp_ms(A,fun(set(product_prod(A,A)),set(product_prod(A,A)))),Uu) = aa(product_prod(A,A),fun(set(product_prod(A,A)),set(product_prod(A,A))),insert2(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Uu),Uu)) ).

% ATP.lambda_16
tff(fact_8200_ATP_Olambda__17,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_adp(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_17
tff(fact_8201_ATP_Olambda__18,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_adq(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_18
tff(fact_8202_ATP_Olambda__19,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A] : aa(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C)),aTP_Lamp_aic(A,fun(product_prod(B,C),product_prod(product_prod(A,B),C))),Uu) = aa(fun(B,fun(C,product_prod(product_prod(A,B),C))),fun(product_prod(B,C),product_prod(product_prod(A,B),C)),product_case_prod(B,C,product_prod(product_prod(A,B),C)),aTP_Lamp_aib(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu)) ).

% ATP.lambda_19
tff(fact_8203_ATP_Olambda__20,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_mv(real,real),Uu) = suminf(real,aTP_Lamp_br(real,fun(nat,real),Uu)) ).

% ATP.lambda_20
tff(fact_8204_ATP_Olambda__21,axiom,
    ! [Uu: nat] : aa(nat,set(nat),aTP_Lamp_wr(nat,set(nat)),Uu) = aa(fun(nat,$o),set(nat),collect(nat),aTP_Lamp_bi(nat,fun(nat,$o),Uu)) ).

% ATP.lambda_21
tff(fact_8205_ATP_Olambda__22,axiom,
    ! [A: $tType,Uu: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_akt(fun(A,$o),$o),Uu)
    <=> aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_az(fun(A,$o),fun(A,$o),Uu))) ) ).

% ATP.lambda_22
tff(fact_8206_ATP_Olambda__23,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real] : aa(real,filter(A),aTP_Lamp_wj(real,filter(A)),Uu) = principal(A,aa(fun(A,$o),set(A),collect(A),aTP_Lamp_wi(real,fun(A,$o),Uu))) ) ).

% ATP.lambda_23
tff(fact_8207_ATP_Olambda__24,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_yv(fun(A,B),set(product_prod(A,B))),Uu) = aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aTP_Lamp_yu(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_24
tff(fact_8208_ATP_Olambda__25,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(real,real)),aTP_Lamp_xg(real,filter(product_prod(real,real))),Uu) = principal(product_prod(real,real),aa(fun(product_prod(real,real),$o),set(product_prod(real,real)),collect(product_prod(real,real)),aa(fun(real,fun(real,$o)),fun(product_prod(real,real),$o),product_case_prod(real,real,$o),aTP_Lamp_xf(real,fun(real,fun(real,$o)),Uu)))) ).

% ATP.lambda_25
tff(fact_8209_ATP_Olambda__26,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(complex,complex)),aTP_Lamp_xe(real,filter(product_prod(complex,complex))),Uu) = principal(product_prod(complex,complex),aa(fun(product_prod(complex,complex),$o),set(product_prod(complex,complex)),collect(product_prod(complex,complex)),aa(fun(complex,fun(complex,$o)),fun(product_prod(complex,complex),$o),product_case_prod(complex,complex,$o),aTP_Lamp_xd(real,fun(complex,fun(complex,$o)),Uu)))) ).

% ATP.lambda_26
tff(fact_8210_ATP_Olambda__27,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real] : aa(real,filter(product_prod(A,A)),aTP_Lamp_xc(real,filter(product_prod(A,A))),Uu) = principal(product_prod(A,A),aa(fun(product_prod(A,A),$o),set(product_prod(A,A)),collect(product_prod(A,A)),aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aTP_Lamp_xb(real,fun(A,fun(A,$o)),Uu)))) ) ).

% ATP.lambda_27
tff(fact_8211_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_qw(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_28
tff(fact_8212_ATP_Olambda__29,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_adn(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(list(B),nat,size_size(list(B)),Uu)) ).

% ATP.lambda_29
tff(fact_8213_ATP_Olambda__30,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_ada(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_30
tff(fact_8214_ATP_Olambda__31,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_gs(nat,fun(nat,product_prod(nat,nat))),Uu) = product_Pair(nat,nat,aa(nat,nat,suc,Uu)) ).

% ATP.lambda_31
tff(fact_8215_ATP_Olambda__32,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_wo(A,filter(A)),Uu) = principal(A,set_ord_atLeast(A,Uu)) ) ).

% ATP.lambda_32
tff(fact_8216_ATP_Olambda__33,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_wp(A,filter(A)),Uu) = principal(A,set_ord_atLeast(A,Uu)) ) ).

% ATP.lambda_33
tff(fact_8217_ATP_Olambda__34,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_wc(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_34
tff(fact_8218_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: A] : aa(A,filter(A),aTP_Lamp_wk(A,filter(A)),Uu) = principal(A,aa(A,set(A),set_ord_atMost(A),Uu)) ) ).

% ATP.lambda_35
tff(fact_8219_ATP_Olambda__36,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( aa(A,$o,aTP_Lamp_abg(A,$o),Uu)
        <=> ? [N: int] :
              ( ( Uu = aa(int,A,ring_1_of_int(A),N) )
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N) ) ) ) ).

% ATP.lambda_36
tff(fact_8220_ATP_Olambda__37,axiom,
    ! [Uu: fun(nat,rat)] :
      ( aa(fun(nat,rat),$o,aTP_Lamp_afp(fun(nat,rat),$o),Uu)
    <=> ? [R5: rat] :
          ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),zero_zero(rat)),R5)
          & ? [K3: nat] :
            ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),N)
             => aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),R5),aa(nat,rat,Uu,N)) ) ) ) ).

% ATP.lambda_37
tff(fact_8221_ATP_Olambda__38,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aTP_Lamp_aeb(product_prod(A,A),$o),Uu)
    <=> ? [X: A] : Uu = aa(A,product_prod(A,A),product_Pair(A,A,X),X) ) ).

% ATP.lambda_38
tff(fact_8222_ATP_Olambda__39,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_abc(real,$o),Uu)
    <=> ? [I4: int,N: nat] :
          ( ( Uu = divide_divide(real,aa(int,real,ring_1_of_int(real),I4),aa(nat,real,semiring_1_of_nat(real),N)) )
          & ( N != zero_zero(nat) ) ) ) ).

% ATP.lambda_39
tff(fact_8223_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zv(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Y4) ) ) ) ).

% ATP.lambda_40
tff(fact_8224_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zu(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),X) ) ) ) ).

% ATP.lambda_41
tff(fact_8225_ATP_Olambda__42,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zx(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y4) ) ) ) ).

% ATP.lambda_42
tff(fact_8226_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( aa(product_prod(A,A),$o,aTP_Lamp_zw(product_prod(A,A),$o),Uu)
        <=> ? [X: A,Y4: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X),Y4) )
              & aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X) ) ) ) ).

% ATP.lambda_43
tff(fact_8227_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_gd(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_44
tff(fact_8228_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_ep(nat,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ).

% ATP.lambda_45
tff(fact_8229_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_ge(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),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,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_46
tff(fact_8230_ATP_Olambda__47,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      aa(nat,extended_enat,aTP_Lamp_akb(extended_enat,fun(nat,extended_enat),Uu),Uua) = extended_case_enat(extended_enat,aTP_Lamp_aka(nat,fun(nat,extended_enat),Uua),
        $ite(Uua = zero_zero(nat),zero_zero(extended_enat),extend4730790105801354508finity(extended_enat)),
        Uu) ).

% ATP.lambda_47
tff(fact_8231_ATP_Olambda__48,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ajc(extended_enat,fun(nat,$o),Uu),Uua)
    <=> extended_case_enat($o,aa(nat,fun(nat,$o),aTP_Lamp_ag(nat,fun(nat,$o)),Uua),$false,Uu) ) ).

% ATP.lambda_48
tff(fact_8232_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_eq(nat,fun(nat,A),Uu),Uua) = $ite(~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua)),zero_zero(A)) ) ).

% ATP.lambda_49
tff(fact_8233_ATP_Olambda__50,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,nat)] :
      ( aa(fun(A,nat),$o,aTP_Lamp_abx(set(A),fun(fun(A,nat),$o),Uu),Uua)
    <=> $ite(aa(set(A),$o,finite_finite2(A),Uu),bij_betw(A,nat,Uua,Uu,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(A),nat,finite_card(A),Uu))),bij_betw(A,nat,Uua,Uu,top_top(set(nat)))) ) ).

% ATP.lambda_50
tff(fact_8234_ATP_Olambda__51,axiom,
    ! [Uu: extended_enat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ajd(extended_enat,fun(nat,$o),Uu),Uua)
    <=> extended_case_enat($o,aa(nat,fun(nat,$o),ord_less(nat),Uua),$true,Uu) ) ).

% ATP.lambda_51
tff(fact_8235_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_xu(fun(A,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),Uua),Uu),top_top(A)) ) ).

% ATP.lambda_52
tff(fact_8236_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_mm(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_53
tff(fact_8237_ATP_Olambda__54,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_aev(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_54
tff(fact_8238_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_xq(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_55
tff(fact_8239_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_aer(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),Uu,Uua),Uua) ) ).

% ATP.lambda_56
tff(fact_8240_ATP_Olambda__57,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_acz(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_acy(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_57
tff(fact_8241_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ia(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hz(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_58
tff(fact_8242_ATP_Olambda__59,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hf(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_he(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_59
tff(fact_8243_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bq(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,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_60
tff(fact_8244_ATP_Olambda__61,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ff(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,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),binomial(Uu,Uua))) ) ).

% ATP.lambda_61
tff(fact_8245_ATP_Olambda__62,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_gh(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_62
tff(fact_8246_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_gg(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uua),divide_divide(real,pi,aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_63
tff(fact_8247_ATP_Olambda__64,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_jj(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Uu)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_64
tff(fact_8248_ATP_Olambda__65,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_hv(nat,fun(nat,a)),Uu),Uua) = $let(
        m2: a,
        m2:= aa(a,a,aa(a,fun(a,a),times_times(a),aa(num,a,numeral_numeral(a),bit0(one2))),aa(nat,a,semiring_1_of_nat(a),Uu)),
        $ite(Uua = zero_zero(nat),m2,aa(a,a,aa(a,fun(a,a),plus_plus(a),m2),one_one(a))) ) ).

% ATP.lambda_65
tff(fact_8249_ATP_Olambda__66,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_abt(complex,fun(real,$o),Uu),Uua)
    <=> ( ( sgn_sgn(complex,Uu) = cis(Uua) )
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi) ) ) ).

% ATP.lambda_66
tff(fact_8250_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_go(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Uua)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less(real),Uu),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_67
tff(fact_8251_ATP_Olambda__68,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_gt(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,aa(rat,fun(rat,$o),ord_less(rat),Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_68
tff(fact_8252_ATP_Olambda__69,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_br(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(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),bit0(one2)))),one_one(nat))))) ).

% ATP.lambda_69
tff(fact_8253_ATP_Olambda__70,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_mw(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_70
tff(fact_8254_ATP_Olambda__71,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_qo(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_71
tff(fact_8255_ATP_Olambda__72,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_fi(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),binomial(Uu,Uua))) ) ).

% ATP.lambda_72
tff(fact_8256_ATP_Olambda__73,axiom,
    ! [Uu: rat,Uua: product_prod(int,int)] :
      ( aa(product_prod(int,int),$o,aTP_Lamp_ahm(rat,fun(product_prod(int,int),$o),Uu),Uua)
    <=> ( ( Uu = fract(aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),Uua))
        & algebr8660921524188924756oprime(int,aa(product_prod(int,int),int,product_fst(int,int),Uua),aa(product_prod(int,int),int,product_snd(int,int),Uua)) ) ) ).

% ATP.lambda_73
tff(fact_8257_ATP_Olambda__74,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_acg(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(A),fun(set(set(A)),set(set(A))),insert2(set(A)),aa(set(A),set(A),image(A,A,Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))))),bot_bot(set(set(A)))) ).

% ATP.lambda_74
tff(fact_8258_ATP_Olambda__75,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dy(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),gbinomial(A,Uu,Uua)),aa(A,A,minus_minus(A,divide_divide(A,Uu,aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_75
tff(fact_8259_ATP_Olambda__76,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aiu(set(A),fun(set(A),$o),Uu),Uua)
        <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uu),Uua)
            & ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = top_top(set(A)) ) ) ) ) ).

% ATP.lambda_76
tff(fact_8260_ATP_Olambda__77,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ahn(nat,fun(nat,$o)),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uu),Uua)
        & ( Uu != Uua ) ) ) ).

% ATP.lambda_77
tff(fact_8261_ATP_Olambda__78,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_iz(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_78
tff(fact_8262_ATP_Olambda__79,axiom,
    ! [Uu: set(int),Uua: int] :
      ( aa(int,$o,aTP_Lamp_aaq(set(int),fun(int,$o),Uu),Uua)
    <=> ( member(int,Uua,Uu)
        & ! [X: int] :
            ( member(int,X,Uu)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X),Uua) ) ) ) ).

% ATP.lambda_79
tff(fact_8263_ATP_Olambda__80,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aar(set(set(A)),fun(set(A),$o),Uu),Uua)
    <=> ( member(set(A),Uua,Uu)
        & ! [X: set(A)] :
            ( member(set(A),X,Uu)
           => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),X) ) ) ) ).

% ATP.lambda_80
tff(fact_8264_ATP_Olambda__81,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(product_prod(A,A))] :
      ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),aTP_Lamp_ago(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o)),Uu),Uua)
    <=> ( aa(set(product_prod(A,A)),$o,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),$o),ord_less_eq(set(product_prod(A,A))),Uu),Uua)
        & ! [A9: A,B7: A,C6: A] :
            ( ( member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A9),B7),Uua)
              & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,B7),C6),Uu) )
           => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,A9),B7),Uu) ) ) ) ).

% ATP.lambda_81
tff(fact_8265_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_em(real,fun(nat,real),Uu),Uua) = divide_divide(real,aa(nat,real,power_power(real,Uu),Uua),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_82
tff(fact_8266_ATP_Olambda__83,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( aa(set(set(A)),$o,aTP_Lamp_afw(set(set(A)),fun(set(set(A)),$o),Uu),Uua)
    <=> ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less_eq(set(set(A))),Uua),Uu)
        & chain_subset(A,Uua) ) ) ).

% ATP.lambda_83
tff(fact_8267_ATP_Olambda__84,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_ji(set(A),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_84
tff(fact_8268_ATP_Olambda__85,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_ie(set(A),fun(set(A),$o)),Uu),Uua)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uu),Uua)
        & aa(set(A),$o,finite_finite2(A),Uua) ) ) ).

% ATP.lambda_85
tff(fact_8269_ATP_Olambda__86,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_xx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,Uu),Uua)),Uua) ).

% ATP.lambda_86
tff(fact_8270_ATP_Olambda__87,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,set(product_prod(B,A)),aTP_Lamp_mk(B,fun(A,set(product_prod(B,A))),Uu),Uua) = aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),product_Pair(B,A,Uu),Uua)),bot_bot(set(product_prod(B,A)))) ).

% ATP.lambda_87
tff(fact_8271_ATP_Olambda__88,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_au(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,power_power(complex,Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_88
tff(fact_8272_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bl(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,power_power(A,Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_89
tff(fact_8273_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bg(A,fun(A,$o),Uu),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu) ) ) ) ).

% ATP.lambda_90
tff(fact_8274_ATP_Olambda__91,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(B,fun(C,$o)),Uua: fun(A,product_prod(B,C))] :
      ( aa(fun(A,product_prod(B,C)),$o,aTP_Lamp_ahk(fun(B,fun(C,$o)),fun(fun(A,product_prod(B,C)),$o),Uu),Uua)
    <=> aa(set(product_prod(B,C)),$o,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),$o),ord_less_eq(set(product_prod(B,C))),aa(set(A),set(product_prod(B,C)),image2(A,product_prod(B,C),Uua),top_top(set(A)))),aa(fun(product_prod(B,C),$o),set(product_prod(B,C)),collect(product_prod(B,C)),aa(fun(B,fun(C,$o)),fun(product_prod(B,C),$o),product_case_prod(B,C,$o),Uu))) ) ).

% ATP.lambda_91
tff(fact_8275_ATP_Olambda__92,axiom,
    ! [A: $tType,Uu: fun(set(A),$o),Uua: set(A)] :
      ( aa(set(A),$o,aa(fun(set(A),$o),fun(set(A),$o),aTP_Lamp_zr(fun(set(A),$o),fun(set(A),$o)),Uu),Uua)
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A8: set(A),A9: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),A9),A8) )
            & aa(set(A),$o,Uu,A8) ) ) ) ).

% ATP.lambda_92
tff(fact_8276_ATP_Olambda__93,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,B)] :
      ( aa(fun(A,B),$o,aTP_Lamp_akp(set(B),fun(fun(A,B),$o),Uu),Uua)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image2(A,B,Uua),top_top(set(A)))),Uu) ) ).

% ATP.lambda_93
tff(fact_8277_ATP_Olambda__94,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(set(A)),aTP_Lamp_acf(set(product_prod(A,A)),fun(A,set(set(A))),Uu),Uua) = equiv_quotient(A,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))),Uu) ).

% ATP.lambda_94
tff(fact_8278_ATP_Olambda__95,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bp(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(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),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_95
tff(fact_8279_ATP_Olambda__96,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_gk(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi)
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_96
tff(fact_8280_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: set(A),Uua: list(A)] :
          ( aa(list(A),$o,aTP_Lamp_ala(set(A),fun(list(A),$o),Uu),Uua)
        <=> ( sorted_wrt(A,ord_less(A),Uua)
            & ( aa(list(A),set(A),set2(A),Uua) = Uu ) ) ) ) ).

% ATP.lambda_97
tff(fact_8281_ATP_Olambda__98,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ya(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ) ).

% ATP.lambda_98
tff(fact_8282_ATP_Olambda__99,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_yf(nat,fun(nat,$o),Uu),Uua)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),zero_zero(nat)),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu)) ) ) ).

% ATP.lambda_99
tff(fact_8283_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_100
tff(fact_8284_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_df(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_101
tff(fact_8285_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ik(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_102
tff(fact_8286_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ih(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_103
tff(fact_8287_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_104
tff(fact_8288_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bs(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,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_105
tff(fact_8289_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ci(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,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_106
tff(fact_8290_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cx(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,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_107
tff(fact_8291_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)))) ) ).

% ATP.lambda_108
tff(fact_8292_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(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_109
tff(fact_8293_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_110
tff(fact_8294_ATP_Olambda__111,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ei(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_111
tff(fact_8295_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_vf(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y4: A] :
                ( aa(A,$o,Uu,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y4) ) ) ) ) ).

% ATP.lambda_112
tff(fact_8296_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_vm(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y4: A] :
                ( aa(A,$o,Uu,Y4)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y4),Uua) ) ) ) ) ).

% ATP.lambda_113
tff(fact_8297_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,A,aTP_Lamp_aex(fun(A,real),fun(A,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(A,real,Uu,Uua)),Uua) ) ).

% ATP.lambda_114
tff(fact_8298_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aes(fun(A,A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,Uu,Uua)),Uua) ) ) ).

% ATP.lambda_115
tff(fact_8299_ATP_Olambda__116,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ue(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_116
tff(fact_8300_ATP_Olambda__117,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,set(A),aTP_Lamp_kd(fun(B,A),fun(B,set(A)),Uu),Uua) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),aa(B,A,Uu,Uua)),bot_bot(set(A))) ).

% ATP.lambda_117
tff(fact_8301_ATP_Olambda__118,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ajb(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_118
tff(fact_8302_ATP_Olambda__119,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_cv(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_119
tff(fact_8303_ATP_Olambda__120,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_hj(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_120
tff(fact_8304_ATP_Olambda__121,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_qp(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_qo(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_121
tff(fact_8305_ATP_Olambda__122,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_rm(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_qo(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_122
tff(fact_8306_ATP_Olambda__123,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: fun(A,real)] : aa(fun(A,real),A,aTP_Lamp_aio(set(A),fun(fun(A,real),A),Uu),Uua) = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),Uua)),Uu) ) ).

% ATP.lambda_123
tff(fact_8307_ATP_Olambda__124,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_rv(fun(A,B),fun(A,real),Uu),Uua) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_124
tff(fact_8308_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aiz(set(A),fun(set(A),$o),Uu),Uua)
        <=> ( ~ real_V358717886546972837endent(A,Uua)
            & ( real_Vector_span(A,Uua) = real_Vector_span(A,Uu) ) ) ) ) ).

% ATP.lambda_125
tff(fact_8309_ATP_Olambda__126,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_fl(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_126
tff(fact_8310_ATP_Olambda__127,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_mr(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Uua)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_127
tff(fact_8311_ATP_Olambda__128,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_rl(A,fun(nat,A),Uu),Uua) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_128
tff(fact_8312_ATP_Olambda__129,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_rk(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,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_129
tff(fact_8313_ATP_Olambda__130,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_eb(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_130
tff(fact_8314_ATP_Olambda__131,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_eo(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_131
tff(fact_8315_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_vp(A,fun(set(A),$o),Uu),Uua)
        <=> ( topolo1002775350975398744n_open(A,Uua)
            & member(A,Uu,Uua) ) ) ) ).

% ATP.lambda_132
tff(fact_8316_ATP_Olambda__133,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aaz(set(A),fun(set(A),$o),Uu),Uua)
    <=> ( aa(set(A),$o,finite_finite2(A),Uua)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ) ).

% ATP.lambda_133
tff(fact_8317_ATP_Olambda__134,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: list(product_prod(A,B))] :
      ( aa(list(product_prod(A,B)),$o,aTP_Lamp_ahp(fun(A,fun(B,$o)),fun(list(product_prod(A,B)),$o),Uu),Uua)
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),Uua)),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),Uu))) ) ).

% ATP.lambda_134
tff(fact_8318_ATP_Olambda__135,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_jp(code_integer,fun(code_integer,num)),Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_135
tff(fact_8319_ATP_Olambda__136,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_jf(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
        l2: nat,
        l2:= code_nat_of_integer(Uu),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_136
tff(fact_8320_ATP_Olambda__137,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_afn(set(A),fun(list(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu) ) ).

% ATP.lambda_137
tff(fact_8321_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_du(nat,fun(nat,A),Uu),Uua) = gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua),Uu) ) ).

% ATP.lambda_138
tff(fact_8322_ATP_Olambda__139,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aTP_Lamp_ahi(set(nat),fun(product_prod(A,nat),$o),Uu),Uua)
    <=> member(nat,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua),Uu) ) ).

% ATP.lambda_139
tff(fact_8323_ATP_Olambda__140,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_abw(set(nat),fun(nat,$o),Uu),Uua)
    <=> member(nat,aa(nat,nat,suc,Uua),Uu) ) ).

% ATP.lambda_140
tff(fact_8324_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gr(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_141
tff(fact_8325_ATP_Olambda__142,axiom,
    ! [A: $tType,Uu: A,Uua: set(set(A))] : aa(set(set(A)),set(set(A)),aa(A,fun(set(set(A)),set(set(A))),aTP_Lamp_lx(A,fun(set(set(A)),set(set(A)))),Uu),Uua) = aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),Uua),aa(set(set(A)),set(set(A)),image2(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu)),Uua)) ).

% ATP.lambda_142
tff(fact_8326_ATP_Olambda__143,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B] : aa(B,set(A),aTP_Lamp_acj(set(product_prod(B,A)),fun(B,set(A)),Uu),Uua) = aa(set(B),set(A),image(B,A,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_143
tff(fact_8327_ATP_Olambda__144,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B] : aa(B,set(A),aTP_Lamp_afu(fun(A,B),fun(B,set(A)),Uu),Uua) = vimage(A,B,Uu,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B)))) ).

% ATP.lambda_144
tff(fact_8328_ATP_Olambda__145,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_gj(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_145
tff(fact_8329_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rg(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_146
tff(fact_8330_ATP_Olambda__147,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ajo(fun(A,A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),aa(A,A,Uu,Uua)) ) ) ).

% ATP.lambda_147
tff(fact_8331_ATP_Olambda__148,axiom,
    ! [Uu: nat,Uua: vEBT_VEBT] : aa(vEBT_VEBT,vEBT_VEBT,aTP_Lamp_ajz(nat,fun(vEBT_VEBT,vEBT_VEBT),Uu),Uua) = vEBT_VEBT_elim_dead(Uua,extended_enat2(aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% ATP.lambda_148
tff(fact_8332_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_io(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_149
tff(fact_8333_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_rh(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_150
tff(fact_8334_ATP_Olambda__151,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ma(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),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ).

% ATP.lambda_151
tff(fact_8335_ATP_Olambda__152,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: A] :
          ( aa(A,$o,aTP_Lamp_wi(real,fun(A,$o),Uu),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu),real_V7770717601297561774m_norm(A,Uua)) ) ) ).

% ATP.lambda_152
tff(fact_8336_ATP_Olambda__153,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ye(set(product_prod(A,A)),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu)) ) ).

% ATP.lambda_153
tff(fact_8337_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_qv(A,fun(nat,A),Uu),Uua) = divide_divide(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_154
tff(fact_8338_ATP_Olambda__155,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_adc(nat,fun(list(A),$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_155
tff(fact_8339_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hu(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_156
tff(fact_8340_ATP_Olambda__157,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ir(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_157
tff(fact_8341_ATP_Olambda__158,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hq(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_158
tff(fact_8342_ATP_Olambda__159,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_adj(list(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_159
tff(fact_8343_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_ail(A,fun(real,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu) ) ).

% ATP.lambda_160
tff(fact_8344_ATP_Olambda__161,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_ac(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ).

% ATP.lambda_161
tff(fact_8345_ATP_Olambda__162,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ag(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_162
tff(fact_8346_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_jg(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_163
tff(fact_8347_ATP_Olambda__164,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajt(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_164
tff(fact_8348_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aii(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_165
tff(fact_8349_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajg(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_166
tff(fact_8350_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ta(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_167
tff(fact_8351_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_er(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_168
tff(fact_8352_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mj(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_169
tff(fact_8353_ATP_Olambda__170,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_jy(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_170
tff(fact_8354_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_yl(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_171
tff(fact_8355_ATP_Olambda__172,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_8356_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_jh(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_173
tff(fact_8357_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_tb(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_174
tff(fact_8358_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aju(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_175
tff(fact_8359_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aij(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_176
tff(fact_8360_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ahy(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_177
tff(fact_8361_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajs(A,fun(A,$o)),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_178
tff(fact_8362_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ef(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),Uu) ) ) ).

% ATP.lambda_179
tff(fact_8363_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_qu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_180
tff(fact_8364_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_kv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_181
tff(fact_8365_ATP_Olambda__182,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_jo(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu) ).

% ATP.lambda_182
tff(fact_8366_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_nb(nat,fun(real,real),Uu),Uua) = aa(nat,real,power_power(real,Uua),Uu) ).

% ATP.lambda_183
tff(fact_8367_ATP_Olambda__184,axiom,
    ! [A: $tType] :
      ( counta4013691401010221786attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_adw(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_184
tff(fact_8368_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_km(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),Uu) ) ).

% ATP.lambda_185
tff(fact_8369_ATP_Olambda__186,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_acu(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_186
tff(fact_8370_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aig(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_187
tff(fact_8371_ATP_Olambda__188,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_nd(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_188
tff(fact_8372_ATP_Olambda__189,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_bi(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_8373_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bc(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),dvd_dvd(A),Uua),Uu) ) ) ).

% ATP.lambda_190
tff(fact_8374_ATP_Olambda__191,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_aih(B,fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uu) ).

% ATP.lambda_191
tff(fact_8375_ATP_Olambda__192,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_agt(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uua),Uu) ).

% ATP.lambda_192
tff(fact_8376_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: int,Uua: A] : aa(A,A,aTP_Lamp_yz(int,fun(A,A),Uu),Uua) = power_int(A,Uua,Uu) ) ).

% ATP.lambda_193
tff(fact_8377_ATP_Olambda__194,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ra(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_194
tff(fact_8378_ATP_Olambda__195,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( aa(B,$o,aTP_Lamp_xy(set(B),fun(B,$o),Uu),Uua)
    <=> member(B,Uua,Uu) ) ).

% ATP.lambda_195
tff(fact_8379_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ys(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_196
tff(fact_8380_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_id(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_197
tff(fact_8381_ATP_Olambda__198,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_xt(set(A),fun(A,$o),Uu),Uua)
        <=> member(A,Uua,Uu) ) ) ).

% ATP.lambda_198
tff(fact_8382_ATP_Olambda__199,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_a(set(A),fun(A,$o)),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_199
tff(fact_8383_ATP_Olambda__200,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_xz(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_200
tff(fact_8384_ATP_Olambda__201,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_add(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_201
tff(fact_8385_ATP_Olambda__202,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ae(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_202
tff(fact_8386_ATP_Olambda__203,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_um(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_203
tff(fact_8387_ATP_Olambda__204,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_uf(fun(A,real),fun(A,$o),Uu),Uua)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_204
tff(fact_8388_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ui(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_205
tff(fact_8389_ATP_Olambda__206,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ud(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_206
tff(fact_8390_ATP_Olambda__207,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agw(nat,fun(nat,set(nat)),Uu),Uua) = order_underS(nat,bNF_Ca8665028551170535155natLeq,Uu) ).

% ATP.lambda_207
tff(fact_8391_ATP_Olambda__208,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_ahg(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image2(product_prod(A,B),B,product_snd(A,B)),Uu) ).

% ATP.lambda_208
tff(fact_8392_ATP_Olambda__209,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_sg(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,divide_divide(A,one_one(A),Uua)) ) ).

% ATP.lambda_209
tff(fact_8393_ATP_Olambda__210,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ic(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_210
tff(fact_8394_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_dd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_211
tff(fact_8395_ATP_Olambda__212,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fo(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_212
tff(fact_8396_ATP_Olambda__213,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_hm(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_213
tff(fact_8397_ATP_Olambda__214,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_de(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_214
tff(fact_8398_ATP_Olambda__215,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_qt(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_215
tff(fact_8399_ATP_Olambda__216,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_abp(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_min(A),Uu),Uua)) ) ).

% ATP.lambda_216
tff(fact_8400_ATP_Olambda__217,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_abd(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_max(A),Uu),Uua)) ) ).

% ATP.lambda_217
tff(fact_8401_ATP_Olambda__218,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_abf(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_218
tff(fact_8402_ATP_Olambda__219,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_abe(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_219
tff(fact_8403_ATP_Olambda__220,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_act(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_acs(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_220
tff(fact_8404_ATP_Olambda__221,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),$o),aa(nat,fun(nat,fun(product_prod(nat,nat),$o)),aTP_Lamp_acp(nat,fun(nat,fun(product_prod(nat,nat),$o))),Uu),Uua) = aa(fun(nat,fun(nat,$o)),fun(product_prod(nat,nat),$o),product_case_prod(nat,nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_aco(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua)) ).

% ATP.lambda_221
tff(fact_8405_ATP_Olambda__222,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_ns(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_nr(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_222
tff(fact_8406_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_ne(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_gb(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_223
tff(fact_8407_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A] : aa(A,set(A),aTP_Lamp_se(real,fun(A,set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(A,fun(A,$o),aTP_Lamp_sd(real,fun(A,fun(A,$o)),Uu),Uua)) ) ).

% ATP.lambda_224
tff(fact_8408_ATP_Olambda__225,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_abq(nat,fun(nat,complex),Uu),Uua) = cis(divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua)),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_225
tff(fact_8409_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_of(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_226
tff(fact_8410_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aey(fun(A,real),fun(A,$o),Uu),Uua)
        <=> ( aa(A,real,Uu,Uua) != zero_zero(real) ) ) ) ).

% ATP.lambda_227
tff(fact_8411_ATP_Olambda__228,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_mi(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)),image2(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_228
tff(fact_8412_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_bo(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_229
tff(fact_8413_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gn(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_230
tff(fact_8414_ATP_Olambda__231,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_adk(list(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,aa(list(A),set(A),set2(A),Uu)) ) ).

% ATP.lambda_231
tff(fact_8415_ATP_Olambda__232,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agl(nat,fun(nat,set(nat)),Uu),Uua) = aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,Uu),Uua)) ).

% ATP.lambda_232
tff(fact_8416_ATP_Olambda__233,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_rf(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_233
tff(fact_8417_ATP_Olambda__234,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(C,product_prod(product_prod(A,B),C)),aTP_Lamp_aib(A,fun(B,fun(C,product_prod(product_prod(A,B),C))),Uu),Uua) = product_Pair(product_prod(A,B),C,aa(B,product_prod(A,B),product_Pair(A,B,Uu),Uua)) ).

% ATP.lambda_234
tff(fact_8418_ATP_Olambda__235,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,extended_enat,aTP_Lamp_aka(nat,fun(nat,extended_enat),Uu),Uua) = extended_enat2(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),Uua)) ).

% ATP.lambda_235
tff(fact_8419_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_vw(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu,Uua)) ) ).

% ATP.lambda_236
tff(fact_8420_ATP_Olambda__237,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_wl(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu)) ) ).

% ATP.lambda_237
tff(fact_8421_ATP_Olambda__238,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_ml(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua) = aa(product_prod(B,A),fun(set(product_prod(B,A)),set(product_prod(B,A))),insert2(product_prod(B,A)),aa(A,product_prod(B,A),product_Pair(B,A,Uu),Uua)) ).

% ATP.lambda_238
tff(fact_8422_ATP_Olambda__239,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B] : aa(B,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_agm(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua) = aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert2(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Uu),Uua)) ).

% ATP.lambda_239
tff(fact_8423_ATP_Olambda__240,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_afi(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_240
tff(fact_8424_ATP_Olambda__241,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_afh(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_241
tff(fact_8425_ATP_Olambda__242,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_afg(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_242
tff(fact_8426_ATP_Olambda__243,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aff(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_243
tff(fact_8427_ATP_Olambda__244,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ba(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_244
tff(fact_8428_ATP_Olambda__245,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ahw(A,fun(A,$o)),Uu),Uua)
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_245
tff(fact_8429_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_sy(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_246
tff(fact_8430_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_so(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_247
tff(fact_8431_ATP_Olambda__248,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_adi(A,fun(A,$o)),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_248
tff(fact_8432_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_aeq(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_lfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_249
tff(fact_8433_ATP_Olambda__250,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(A,fun(A,A)),Uua: A] : aa(A,A,aTP_Lamp_ajn(fun(A,fun(A,A)),fun(A,A),Uu),Uua) = complete_lattice_gfp(A,aa(A,fun(A,A),Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_8434_ATP_Olambda__251,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_cz(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_8435_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_dh(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_8436_ATP_Olambda__253,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_cq(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_8437_ATP_Olambda__254,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_gx(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_254
tff(fact_8438_ATP_Olambda__255,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_ph(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_255
tff(fact_8439_ATP_Olambda__256,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: A] : aa(A,B,aTP_Lamp_cj(fun(A,$o),fun(A,B),Uu),Uua) = aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uua)) ) ).

% ATP.lambda_256
tff(fact_8440_ATP_Olambda__257,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_qz(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_257
tff(fact_8441_ATP_Olambda__258,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_od(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_258
tff(fact_8442_ATP_Olambda__259,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_tt(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_259
tff(fact_8443_ATP_Olambda__260,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ux(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_260
tff(fact_8444_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_mz(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_261
tff(fact_8445_ATP_Olambda__262,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_qg(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_262
tff(fact_8446_ATP_Olambda__263,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ur(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_263
tff(fact_8447_ATP_Olambda__264,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_pl(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_264
tff(fact_8448_ATP_Olambda__265,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_og(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_265
tff(fact_8449_ATP_Olambda__266,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_hy(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_266
tff(fact_8450_ATP_Olambda__267,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_di(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_267
tff(fact_8451_ATP_Olambda__268,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_268
tff(fact_8452_ATP_Olambda__269,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_nt(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_269
tff(fact_8453_ATP_Olambda__270,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_vk(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_270
tff(fact_8454_ATP_Olambda__271,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_ve(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_271
tff(fact_8455_ATP_Olambda__272,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_sc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_272
tff(fact_8456_ATP_Olambda__273,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pe(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_273
tff(fact_8457_ATP_Olambda__274,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_nv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_274
tff(fact_8458_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_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,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8459_ATP_Olambda__276,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_uy(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8460_ATP_Olambda__277,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_qh(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_277
tff(fact_8461_ATP_Olambda__278,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_pk(fun(A,B),fun(A,B),Uu),Uua) = sgn_sgn(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8462_ATP_Olambda__279,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_da(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_279
tff(fact_8463_ATP_Olambda__280,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_cb(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_280
tff(fact_8464_ATP_Olambda__281,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_vc(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_281
tff(fact_8465_ATP_Olambda__282,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ql(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_282
tff(fact_8466_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ni(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_283
tff(fact_8467_ATP_Olambda__284,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_pv(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_284
tff(fact_8468_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & 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,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_285
tff(fact_8469_ATP_Olambda__286,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_hk(fun(A,B),fun(A,B),Uu),Uua) = exp(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_286
tff(fact_8470_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_pw(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8471_ATP_Olambda__288,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: C] : aa(C,set(product_prod(A,B)),aTP_Lamp_ahq(fun(C,set(product_prod(B,A))),fun(C,set(product_prod(A,B))),Uu),Uua) = converse(B,A,aa(C,set(product_prod(B,A)),Uu,Uua)) ).

% ATP.lambda_288
tff(fact_8472_ATP_Olambda__289,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_wg(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_289
tff(fact_8473_ATP_Olambda__290,axiom,
    ! [D: $tType,C: $tType,Uu: fun(C,set(D)),Uua: C] : aa(C,filter(D),aTP_Lamp_wa(fun(C,set(D)),fun(C,filter(D)),Uu),Uua) = principal(D,aa(C,set(D),Uu,Uua)) ).

% ATP.lambda_290
tff(fact_8474_ATP_Olambda__291,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,filter(A),aTP_Lamp_vq(fun(B,set(A)),fun(B,filter(A)),Uu),Uua) = principal(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_291
tff(fact_8475_ATP_Olambda__292,axiom,
    ! [E5: $tType,A: $tType,Uu: fun(A,set(E5)),Uua: A] : aa(A,filter(E5),aTP_Lamp_vz(fun(A,set(E5)),fun(A,filter(E5)),Uu),Uua) = principal(E5,aa(A,set(E5),Uu,Uua)) ).

% ATP.lambda_292
tff(fact_8476_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_wh(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_293
tff(fact_8477_ATP_Olambda__294,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_lv(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_294
tff(fact_8478_ATP_Olambda__295,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_ot(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_295
tff(fact_8479_ATP_Olambda__296,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_qm(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_8480_ATP_Olambda__297,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_oq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_8481_ATP_Olambda__298,axiom,
    ! [A: $tType,Uu: fun(set(A),fun(A,$o)),Uua: set(A)] : aa(set(A),set(A),aTP_Lamp_ajp(fun(set(A),fun(A,$o)),fun(set(A),set(A)),Uu),Uua) = aa(fun(A,$o),set(A),collect(A),aa(set(A),fun(A,$o),Uu,Uua)) ).

% ATP.lambda_298
tff(fact_8482_ATP_Olambda__299,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_ly(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = aa(B,fun(set(B),set(B)),insert2(B),aa(A,B,Uu,Uua)) ).

% ATP.lambda_299
tff(fact_8483_ATP_Olambda__300,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_lk(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow2(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_300
tff(fact_8484_ATP_Olambda__301,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: B] :
      ( aa(B,$o,aTP_Lamp_jx(fun(B,$o),fun(B,$o),Uu),Uua)
    <=> ~ aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_301
tff(fact_8485_ATP_Olambda__302,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_az(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_302
tff(fact_8486_ATP_Olambda__303,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_uu(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
        <=> ! [X_13: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_13) ) ) ).

% ATP.lambda_303
tff(fact_8487_ATP_Olambda__304,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aki(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ! [X_13: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_13) ) ).

% ATP.lambda_304
tff(fact_8488_ATP_Olambda__305,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_zm(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X_13: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_13) ) ).

% ATP.lambda_305
tff(fact_8489_ATP_Olambda__306,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_agv(nat,fun(nat,set(nat)),Uu),Uua) = aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_af(nat,fun(nat,$o)),Uu)) ).

% ATP.lambda_306
tff(fact_8490_ATP_Olambda__307,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_aay(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),aa(fun(set(A),$o),set(set(A)),collect(set(A)),aa(set(A),fun(set(A),$o),aTP_Lamp_aax(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua))) ).

% ATP.lambda_307
tff(fact_8491_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real] : aa(real,filter(A),aTP_Lamp_vu(A,fun(real,filter(A)),Uu),Uua) = principal(A,aa(fun(A,$o),set(A),collect(A),aa(real,fun(A,$o),aTP_Lamp_vt(A,fun(real,fun(A,$o)),Uu),Uua))) ) ).

% ATP.lambda_308
tff(fact_8492_ATP_Olambda__309,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o)] : aa(fun(B,$o),filter(product_prod(A,B)),aa(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B))),aTP_Lamp_zh(fun(A,$o),fun(fun(B,$o),filter(product_prod(A,B)))),Uu),Uua) = principal(product_prod(A,B),aa(fun(product_prod(A,B),$o),set(product_prod(A,B)),collect(product_prod(A,B)),aa(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),product_case_prod(A,B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_zg(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua)))) ).

% ATP.lambda_309
tff(fact_8493_ATP_Olambda__310,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_akd(fun(B,fun(A,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X: B] : aa(A,$o,aa(B,fun(A,$o),Uu,X),Uua) ) ).

% ATP.lambda_310
tff(fact_8494_ATP_Olambda__311,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aip(set(A),fun(A,$o),Uu),Uua)
        <=> ? [F7: fun(A,real)] :
              ( ( Uua = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),F7)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F7))) )
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F7))),Uu)
              & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F7))) ) ) ) ).

% ATP.lambda_311
tff(fact_8495_ATP_Olambda__312,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aiq(set(A),fun(A,$o),Uu),Uua)
        <=> ? [F7: fun(A,real)] :
              ( ( Uua = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),F7)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F7))) )
              & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),F7)))
              & ! [V6: A] :
                  ( ( aa(A,real,F7,V6) != zero_zero(real) )
                 => member(A,V6,Uu) ) ) ) ) ).

% ATP.lambda_312
tff(fact_8496_ATP_Olambda__313,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aac(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I4: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_313
tff(fact_8497_ATP_Olambda__314,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aal(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X: set(A)] :
                  ( member(set(A),X,Uu)
                 => member(A,aa(set(A),A,F7,X),X) ) ) ) ) ).

% ATP.lambda_314
tff(fact_8498_ATP_Olambda__315,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aak(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X: set(A)] :
                  ( member(set(A),X,Uu)
                 => member(A,aa(set(A),A,F7,X),X) ) ) ) ) ).

% ATP.lambda_315
tff(fact_8499_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( aa(set(A),$o,aTP_Lamp_aam(set(set(A)),fun(set(A),$o),Uu),Uua)
        <=> ? [F7: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image2(set(A),A,F7),Uu) )
              & ! [X: set(A)] :
                  ( member(set(A),X,Uu)
                 => member(A,aa(set(A),A,F7,X),X) ) ) ) ) ).

% ATP.lambda_316
tff(fact_8500_ATP_Olambda__317,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_aao(set(A),fun(set(A),$o),Uu),Uua)
    <=> ? [B11: set(A)] :
          ( ( Uua = aa(set(A),set(A),uminus_uminus(set(A)),B11) )
          & member(set(A),Uu,pow2(A,B11)) ) ) ).

% ATP.lambda_317
tff(fact_8501_ATP_Olambda__318,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( aa(filter(A),$o,aTP_Lamp_aas(set(filter(A)),fun(filter(A),$o),Uu),Uua)
    <=> ! [X: filter(A)] :
          ( member(filter(A),X,Uu)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),Uua),X) ) ) ).

% ATP.lambda_318
tff(fact_8502_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aah(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( member(A,X,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X) ) ) ) ).

% ATP.lambda_319
tff(fact_8503_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aab(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( member(A,X,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),X) ) ) ) ).

% ATP.lambda_320
tff(fact_8504_ATP_Olambda__321,axiom,
    ! [Uu: set(real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_aap(set(real),fun(real,$o),Uu),Uua)
    <=> ! [X: real] :
          ( member(real,X,Uu)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X),Uua) ) ) ).

% ATP.lambda_321
tff(fact_8505_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aai(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( member(A,X,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Uua) ) ) ) ).

% ATP.lambda_322
tff(fact_8506_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aaa(set(A),fun(A,$o),Uu),Uua)
        <=> ! [X: A] :
              ( member(A,X,Uu)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Uua) ) ) ) ).

% ATP.lambda_323
tff(fact_8507_ATP_Olambda__324,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_abi(set(filter(A)),fun(fun(A,$o),$o),Uu),Uua)
    <=> ! [X: filter(A)] :
          ( member(filter(A),X,Uu)
         => aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uua),X) ) ) ).

% ATP.lambda_324
tff(fact_8508_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_vb(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ! [Y4: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y4)
             => aa(A,$o,Uu,Y4) ) ) ) ).

% ATP.lambda_325
tff(fact_8509_ATP_Olambda__326,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_akf(set(product_prod(A,B)),fun(A,$o),Uu),Uua)
    <=> ? [Y4: B] : member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Uua),Y4),Uu) ) ).

% ATP.lambda_326
tff(fact_8510_ATP_Olambda__327,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_zy(fun(B,A),fun(A,$o),Uu),Uua)
    <=> ? [X: B] : Uua = aa(B,A,Uu,X) ) ).

% ATP.lambda_327
tff(fact_8511_ATP_Olambda__328,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_air(set(A),fun(A,$o),Uu),Uua)
        <=> ? [T2: set(A),R5: fun(A,real)] :
              ( ( Uua = aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),R5)),T2) )
              & aa(set(A),$o,finite_finite2(A),T2)
              & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T2),Uu) ) ) ) ).

% ATP.lambda_328
tff(fact_8512_ATP_Olambda__329,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: product_prod(set(A),set(A))] :
      ( aa(product_prod(set(A),set(A)),$o,aTP_Lamp_aeo(set(product_prod(A,A)),fun(product_prod(set(A),set(A)),$o),Uu),Uua)
    <=> ? [X8: set(A),Y5: set(A)] :
          ( ( Uua = aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),X8),Y5) )
          & ( X8 != bot_bot(set(A)) )
          & ! [X: A] :
              ( member(A,X,Y5)
             => ? [Xa3: A] :
                  ( member(A,Xa3,X8)
                  & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Xa3),X),Uu) ) ) ) ) ).

% ATP.lambda_329
tff(fact_8513_ATP_Olambda__330,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: A] : aa(A,set(B),aTP_Lamp_ako(set(product_prod(B,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(B,B)),set(B),field2(B),Uu) ).

% ATP.lambda_330
tff(fact_8514_ATP_Olambda__331,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,A)),Uua: B] : aa(B,set(A),aTP_Lamp_akn(set(product_prod(A,A)),fun(B,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ).

% ATP.lambda_331
tff(fact_8515_ATP_Olambda__332,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A] : aa(A,set(A),aTP_Lamp_agh(set(product_prod(A,A)),fun(A,set(A)),Uu),Uua) = aa(set(product_prod(A,A)),set(A),field2(A),Uu) ).

% ATP.lambda_332
tff(fact_8516_ATP_Olambda__333,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(set(A),set(A)),aa(A,fun(list(A),fun(set(A),set(A))),aTP_Lamp_afj(A,fun(list(A),fun(set(A),set(A)))),Uu),Uua) = aa(A,fun(set(A),set(A)),insert2(A),Uu) ).

% ATP.lambda_333
tff(fact_8517_ATP_Olambda__334,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_age(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_334
tff(fact_8518_ATP_Olambda__335,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_iv(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),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),bit0(one2))),Uua)),one_one(code_integer))),aa(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),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_335
tff(fact_8519_ATP_Olambda__336,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_gp(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),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),bit0(one2))),Uua)),one_one(nat))),aa(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),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_336
tff(fact_8520_ATP_Olambda__337,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_gq(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),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),bit0(one2))),Uua)),one_one(int))),aa(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),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_337
tff(fact_8521_ATP_Olambda__338,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_gm(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),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),bit0(one2))),Uua)),one_one(A))),aa(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),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_338
tff(fact_8522_ATP_Olambda__339,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_cf(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(member(nat,Uub,Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_339
tff(fact_8523_ATP_Olambda__340,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_cu(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_340
tff(fact_8524_ATP_Olambda__341,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: set(A),Uub: A] :
          aa(A,B,aa(set(A),fun(A,B),aTP_Lamp_hi(fun(A,B),fun(set(A),fun(A,B)),Uu),Uua),Uub) = $ite(member(A,Uub,Uua),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_341
tff(fact_8525_ATP_Olambda__342,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ca(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_342
tff(fact_8526_ATP_Olambda__343,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gv(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_343
tff(fact_8527_ATP_Olambda__344,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_by(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_344
tff(fact_8528_ATP_Olambda__345,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bz(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_345
tff(fact_8529_ATP_Olambda__346,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_gw(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_346
tff(fact_8530_ATP_Olambda__347,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_jc(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),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,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_347
tff(fact_8531_ATP_Olambda__348,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_jb(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),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,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uu),Uub))) ).

% ATP.lambda_348
tff(fact_8532_ATP_Olambda__349,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_lz(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),Uub),Uub) ).

% ATP.lambda_349
tff(fact_8533_ATP_Olambda__350,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ce(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(aa(nat,$o,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_350
tff(fact_8534_ATP_Olambda__351,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
          aa(B,A,aa(fun(B,$o),fun(B,A),aTP_Lamp_adh(fun(B,A),fun(fun(B,$o),fun(B,A)),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(B,A,Uu,Uub),zero_zero(A)) ) ).

% ATP.lambda_351
tff(fact_8535_ATP_Olambda__352,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_cr(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_352
tff(fact_8536_ATP_Olambda__353,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_hc(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_353
tff(fact_8537_ATP_Olambda__354,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A,Uub: set(product_prod(A,B))] : aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(A,fun(set(product_prod(A,B)),set(product_prod(A,B))),aTP_Lamp_agn(set(B),fun(A,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uu),Uua),Uub) = finite_fold(B,set(product_prod(A,B)),aTP_Lamp_agm(A,fun(B,fun(set(product_prod(A,B)),set(product_prod(A,B)))),Uua),Uub,Uu) ).

% ATP.lambda_354
tff(fact_8538_ATP_Olambda__355,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B,Uub: set(product_prod(B,A))] : aa(set(product_prod(B,A)),set(product_prod(B,A)),aa(B,fun(set(product_prod(B,A)),set(product_prod(B,A))),aTP_Lamp_xo(set(A),fun(B,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uu),Uua),Uub) = finite_fold(A,set(product_prod(B,A)),aTP_Lamp_ml(B,fun(A,fun(set(product_prod(B,A)),set(product_prod(B,A)))),Uua),Uub,Uu) ).

% ATP.lambda_355
tff(fact_8539_ATP_Olambda__356,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_yw(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_356
tff(fact_8540_ATP_Olambda__357,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: $o] :
      aa($o,set(A),aa(set(A),fun($o,set(A)),aTP_Lamp_mq(set(A),fun(set(A),fun($o,set(A))),Uu),Uua),(Uub)) = $ite((Uub),Uu,Uua) ).

% ATP.lambda_357
tff(fact_8541_ATP_Olambda__358,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_hz(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,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_358
tff(fact_8542_ATP_Olambda__359,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_he(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,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_359
tff(fact_8543_ATP_Olambda__360,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_zn(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_360
tff(fact_8544_ATP_Olambda__361,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_os(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_361
tff(fact_8545_ATP_Olambda__362,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_il(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_362
tff(fact_8546_ATP_Olambda__363,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_ii(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_363
tff(fact_8547_ATP_Olambda__364,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ut(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ) ).

% ATP.lambda_364
tff(fact_8548_ATP_Olambda__365,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_zl(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_365
tff(fact_8549_ATP_Olambda__366,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_pq(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_366
tff(fact_8550_ATP_Olambda__367,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_te(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_367
tff(fact_8551_ATP_Olambda__368,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_ok(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_368
tff(fact_8552_ATP_Olambda__369,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_gz(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_369
tff(fact_8553_ATP_Olambda__370,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_co(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_370
tff(fact_8554_ATP_Olambda__371,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_fy(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_fx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_371
tff(fact_8555_ATP_Olambda__372,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_fw(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_fv(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_372
tff(fact_8556_ATP_Olambda__373,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_fu(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_ft(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_373
tff(fact_8557_ATP_Olambda__374,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_fq(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu,one_one(A),zero_zero(A))),
            aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_374
tff(fact_8558_ATP_Olambda__375,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_ix(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        product_Pair(code_integer,$o,
          $ite(aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_375
tff(fact_8559_ATP_Olambda__376,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aiw(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_376
tff(fact_8560_ATP_Olambda__377,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A)),Uub: set(set(A))] :
      ( aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),aTP_Lamp_ake(set(set(A)),fun(set(set(A)),fun(set(set(A)),$o)),Uu),Uua),Uub)
    <=> ( aa(set(set(A)),$o,pred_chain(set(A),Uu,ord_less(set(A))),Uub)
        & aa(set(set(A)),$o,aa(set(set(A)),fun(set(set(A)),$o),ord_less(set(set(A))),Uua),Uub) ) ) ).

% ATP.lambda_377
tff(fact_8561_ATP_Olambda__378,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajx(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu,Uua),Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_378
tff(fact_8562_ATP_Olambda__379,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_im(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_il(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_379
tff(fact_8563_ATP_Olambda__380,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_ij(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_ii(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_380
tff(fact_8564_ATP_Olambda__381,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_afc(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_afb(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_381
tff(fact_8565_ATP_Olambda__382,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_afa(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,aa(fun(int,fun(int,$o)),fun(product_prod(int,int),$o),product_case_prod(int,int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_aez(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_382
tff(fact_8566_ATP_Olambda__383,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ol(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_ok(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_383
tff(fact_8567_ATP_Olambda__384,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_tf(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aTP_Lamp_te(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_384
tff(fact_8568_ATP_Olambda__385,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: filter(B),Uub: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_aif(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,$o),fun(B,$o),aTP_Lamp_aie(fun(B,A),fun(fun(A,$o),fun(B,$o)),Uu),Uub)),Uua) ) ).

% ATP.lambda_385
tff(fact_8569_ATP_Olambda__386,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_nj(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(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,power_power(real,Uua),Uub)) ).

% ATP.lambda_386
tff(fact_8570_ATP_Olambda__387,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_nk(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real)),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_387
tff(fact_8571_ATP_Olambda__388,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_ek(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A)),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ) ).

% ATP.lambda_388
tff(fact_8572_ATP_Olambda__389,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_ev(fun(nat,A),fun(nat,fun(A,$o)),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_es(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_389
tff(fact_8573_ATP_Olambda__390,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_qi(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(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_390
tff(fact_8574_ATP_Olambda__391,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_qd(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(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_391
tff(fact_8575_ATP_Olambda__392,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_ck(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,zero_zero(A)),Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_392
tff(fact_8576_ATP_Olambda__393,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_qe(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua)),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_393
tff(fact_8577_ATP_Olambda__394,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_nq(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,power_power(real,Uua),Uub)) ).

% ATP.lambda_394
tff(fact_8578_ATP_Olambda__395,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_el(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,Uua,Uub),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_395
tff(fact_8579_ATP_Olambda__396,axiom,
    ! [A: $tType,Uu: fun(set(A),set(A)),Uua: set(A),Uub: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aTP_Lamp_ajq(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),Uu),Uua),Uub) = 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),Uu,Uub)),Uua)),complete_lattice_gfp(set(A),Uu)) ).

% ATP.lambda_396
tff(fact_8580_ATP_Olambda__397,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(set(A),set(A)),Uub: set(A)] : aa(set(A),set(A),aa(fun(set(A),set(A)),fun(set(A),set(A)),aTP_Lamp_ajl(set(A),fun(fun(set(A),set(A)),fun(set(A),set(A))),Uu),Uua),Uub) = 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),Uua,Uub)),Uu)),complete_lattice_gfp(set(A),Uua)) ).

% ATP.lambda_397
tff(fact_8581_ATP_Olambda__398,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_acm(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,minus_minus(nat,Uu),one_one(nat)) )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_398
tff(fact_8582_ATP_Olambda__399,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_ga(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,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_399
tff(fact_8583_ATP_Olambda__400,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_aau(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Uua),Uub),lex(A,Uu)) ) ) ) ).

% ATP.lambda_400
tff(fact_8584_ATP_Olambda__401,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_acn(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_401
tff(fact_8585_ATP_Olambda__402,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(A),fun(list(A),$o),aTP_Lamp_hx(nat,fun(set(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua) ) ) ).

% ATP.lambda_402
tff(fact_8586_ATP_Olambda__403,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_hw(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu) ) ) ).

% ATP.lambda_403
tff(fact_8587_ATP_Olambda__404,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ags(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = aa(nat,nat,suc,Uua) ) ) ) ).

% ATP.lambda_404
tff(fact_8588_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_jk(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_405
tff(fact_8589_ATP_Olambda__406,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_bn(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_406
tff(fact_8590_ATP_Olambda__407,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_bm(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_407
tff(fact_8591_ATP_Olambda__408,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( aa(list(nat),$o,aa(nat,fun(list(nat),$o),aTP_Lamp_acl(nat,fun(nat,fun(list(nat),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(list(nat),nat,groups8242544230860333062m_list(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_408
tff(fact_8592_ATP_Olambda__409,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,option(B))] :
      ( aa(fun(A,option(B)),$o,aa(set(B),fun(fun(A,option(B)),$o),aTP_Lamp_acc(set(A),fun(set(B),fun(fun(A,option(B)),$o)),Uu),Uua),Uub)
    <=> ( ( dom(A,B,Uub) = Uu )
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),ran(A,B,Uub)),Uua) ) ) ).

% ATP.lambda_409
tff(fact_8593_ATP_Olambda__410,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_nf(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_410
tff(fact_8594_ATP_Olambda__411,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bk(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,aa(nat,nat,suc,Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_411
tff(fact_8595_ATP_Olambda__412,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( aa(product_prod(A,nat),$o,aa(nat,fun(product_prod(A,nat),$o),aTP_Lamp_ahj(set(nat),fun(nat,fun(product_prod(A,nat),$o)),Uu),Uua),Uub)
    <=> member(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uub)),Uua),Uu) ) ).

% ATP.lambda_412
tff(fact_8596_ATP_Olambda__413,axiom,
    ! [Uu: nat,Uua: nat,Uub: set(nat)] :
      ( aa(set(nat),$o,aa(nat,fun(set(nat),$o),aTP_Lamp_iw(nat,fun(nat,fun(set(nat),$o)),Uu),Uua),Uub)
    <=> ( member(set(nat),Uub,pow2(nat,set_or7035219750837199246ssThan(nat,zero_zero(nat),Uu)))
        & ( aa(set(nat),nat,finite_card(nat),Uub) = Uua ) ) ) ).

% ATP.lambda_413
tff(fact_8597_ATP_Olambda__414,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_aaw(list(A),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu))
        & member(nat,Uub,Uua) ) ) ).

% ATP.lambda_414
tff(fact_8598_ATP_Olambda__415,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_adl(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua))
        & aa(A,$o,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ) ).

% ATP.lambda_415
tff(fact_8599_ATP_Olambda__416,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_acx(list(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uu))
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_416
tff(fact_8600_ATP_Olambda__417,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_iq(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(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,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_417
tff(fact_8601_ATP_Olambda__418,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aea(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uua),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uub) ) ) ).

% ATP.lambda_418
tff(fact_8602_ATP_Olambda__419,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [Uu: set(A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_ib(set(A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(nat,A,infini527867602293511546merate(A,Uu),Uua)),Uub) ) ) ) ).

% ATP.lambda_419
tff(fact_8603_ATP_Olambda__420,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_abv(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image2(A,B,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_420
tff(fact_8604_ATP_Olambda__421,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_akg(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( order_ofilter(A,Uu,Uua)
        & ( Uua != aa(set(product_prod(A,A)),set(A),field2(A),Uu) )
        & order_ofilter(A,Uu,Uub)
        & ( Uub != aa(set(product_prod(A,A)),set(A),field2(A),Uu) )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),Uub) ) ) ).

% ATP.lambda_421
tff(fact_8605_ATP_Olambda__422,axiom,
    ! [Uu: set(nat),Uua: set(nat),Uub: nat] :
      ( aa(nat,$o,aa(set(nat),fun(nat,$o),aTP_Lamp_aem(set(nat),fun(set(nat),fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & member(nat,aa(set(nat),nat,finite_card(nat),aa(fun(nat,$o),set(nat),collect(nat),aa(nat,fun(nat,$o),aTP_Lamp_ael(set(nat),fun(nat,fun(nat,$o)),Uu),Uub))),Uua) ) ) ).

% ATP.lambda_422
tff(fact_8606_ATP_Olambda__423,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( aa(set(A),$o,aa(nat,fun(set(A),$o),aTP_Lamp_bh(set(A),fun(nat,fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu)
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_423
tff(fact_8607_ATP_Olambda__424,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bj(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),aa(nat,nat,suc,Uua)) ) ) ).

% ATP.lambda_424
tff(fact_8608_ATP_Olambda__425,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_fz(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_425
tff(fact_8609_ATP_Olambda__426,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_nc(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_426
tff(fact_8610_ATP_Olambda__427,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_gc(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,power_power(A,Uu),Uub)) ) ).

% ATP.lambda_427
tff(fact_8611_ATP_Olambda__428,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_eu(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = binomial(aa(nat,nat,minus_minus(nat,Uua),Uub),aa(nat,nat,minus_minus(nat,Uu),Uub)) ).

% ATP.lambda_428
tff(fact_8612_ATP_Olambda__429,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aq(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_429
tff(fact_8613_ATP_Olambda__430,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_as(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_430
tff(fact_8614_ATP_Olambda__431,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ig(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_431
tff(fact_8615_ATP_Olambda__432,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ah(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_432
tff(fact_8616_ATP_Olambda__433,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_if(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uua),Uub) ) ) ).

% ATP.lambda_433
tff(fact_8617_ATP_Olambda__434,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ak(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_434
tff(fact_8618_ATP_Olambda__435,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_aj(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_435
tff(fact_8619_ATP_Olambda__436,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ai(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less(int),Uub),Uua) ) ) ).

% ATP.lambda_436
tff(fact_8620_ATP_Olambda__437,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_aen(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uua),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu) ) ) ).

% ATP.lambda_437
tff(fact_8621_ATP_Olambda__438,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aho(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uua)
        & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),Uub),Uu) ) ) ).

% ATP.lambda_438
tff(fact_8622_ATP_Olambda__439,axiom,
    ! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_abk(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uu)
        & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uua) ) ) ).

% ATP.lambda_439
tff(fact_8623_ATP_Olambda__440,axiom,
    ! [A: $tType,Uu: set(A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afo(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uua,Uu)
        & member(A,Uub,Uu) ) ) ).

% ATP.lambda_440
tff(fact_8624_ATP_Olambda__441,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ael(set(nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( member(nat,Uub,Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_441
tff(fact_8625_ATP_Olambda__442,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ay(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & member(A,Uub,Uua) ) ) ).

% ATP.lambda_442
tff(fact_8626_ATP_Olambda__443,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ajf(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uu )
        & ( Uub = Uu ) ) ) ).

% ATP.lambda_443
tff(fact_8627_ATP_Olambda__444,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_vg(set(A),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),zero_zero(real)),aa(A,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_444
tff(fact_8628_ATP_Olambda__445,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_acq(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_445
tff(fact_8629_ATP_Olambda__446,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_vx(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uua)
       => aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_446
tff(fact_8630_ATP_Olambda__447,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aku(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_447
tff(fact_8631_ATP_Olambda__448,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akv(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_448
tff(fact_8632_ATP_Olambda__449,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bd(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_449
tff(fact_8633_ATP_Olambda__450,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_me(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_450
tff(fact_8634_ATP_Olambda__451,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_an(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_451
tff(fact_8635_ATP_Olambda__452,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_mu(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uua = Uub )
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_452
tff(fact_8636_ATP_Olambda__453,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_am(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_453
tff(fact_8637_ATP_Olambda__454,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,set(B)),Uub: A] :
      ( aa(A,$o,aa(fun(A,set(B)),fun(A,$o),aTP_Lamp_agg(set(A),fun(fun(A,set(B)),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ( aa(A,set(B),Uua,Uub) != bot_bot(set(B)) ) ) ) ).

% ATP.lambda_454
tff(fact_8638_ATP_Olambda__455,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_dz(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_455
tff(fact_8639_ATP_Olambda__456,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_gu(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_456
tff(fact_8640_ATP_Olambda__457,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bu(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_457
tff(fact_8641_ATP_Olambda__458,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_hb(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_458
tff(fact_8642_ATP_Olambda__459,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_jm(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_459
tff(fact_8643_ATP_Olambda__460,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ds(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,Uua,Uub)) ) ) ) ).

% ATP.lambda_460
tff(fact_8644_ATP_Olambda__461,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aw(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_461
tff(fact_8645_ATP_Olambda__462,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ww(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_462
tff(fact_8646_ATP_Olambda__463,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_hd(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_463
tff(fact_8647_ATP_Olambda__464,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_ry(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_464
tff(fact_8648_ATP_Olambda__465,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_xd(real,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(complex,Uua,Uub)),Uu) ) ).

% ATP.lambda_465
tff(fact_8649_ATP_Olambda__466,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_xf(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(real,Uua,Uub)),Uu) ) ).

% ATP.lambda_466
tff(fact_8650_ATP_Olambda__467,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_xb(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_467
tff(fact_8651_ATP_Olambda__468,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_sd(real,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu) ) ) ).

% ATP.lambda_468
tff(fact_8652_ATP_Olambda__469,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_vt(A,fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu)),Uua) ) ) ).

% ATP.lambda_469
tff(fact_8653_ATP_Olambda__470,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_hh(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_470
tff(fact_8654_ATP_Olambda__471,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_kz(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_471
tff(fact_8655_ATP_Olambda__472,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_kx(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_472
tff(fact_8656_ATP_Olambda__473,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ky(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_473
tff(fact_8657_ATP_Olambda__474,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_kw(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_474
tff(fact_8658_ATP_Olambda__475,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aci(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,Uua),Uub),Uu) ) ).

% ATP.lambda_475
tff(fact_8659_ATP_Olambda__476,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(A,B)),fun(A,fun(B,$o)),aTP_Lamp_al(set(product_prod(A,B)),fun(A,fun(B,$o))),Uu),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uub),Uu) ) ).

% ATP.lambda_476
tff(fact_8660_ATP_Olambda__477,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ahx(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Uua),Uub),Uu) ) ).

% ATP.lambda_477
tff(fact_8661_ATP_Olambda__478,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ahr(set(product_prod(B,A)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,Uub),Uua),Uu) ) ).

% ATP.lambda_478
tff(fact_8662_ATP_Olambda__479,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ahv(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Uub),Uua),Uu) ) ).

% ATP.lambda_479
tff(fact_8663_ATP_Olambda__480,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_acy(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_480
tff(fact_8664_ATP_Olambda__481,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_ad(nat,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_481
tff(fact_8665_ATP_Olambda__482,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_av(complex,fun(nat,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,power_power(complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_482
tff(fact_8666_ATP_Olambda__483,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bf(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_483
tff(fact_8667_ATP_Olambda__484,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,A),Uub: A] :
      ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_aid(fun(A,B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uua,aa(A,B,Uu,Uub)) = Uub ) ) ).

% ATP.lambda_484
tff(fact_8668_ATP_Olambda__485,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_dm(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_485
tff(fact_8669_ATP_Olambda__486,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(fun(B,$o),fun(product_prod(A,B),$o),aTP_Lamp_ahh(fun(A,$o),fun(fun(B,$o),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uub))
        & aa(B,$o,Uua,aa(product_prod(A,B),B,product_snd(A,B),Uub)) ) ) ).

% ATP.lambda_486
tff(fact_8670_ATP_Olambda__487,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_nr(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,power_power(real,Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_487
tff(fact_8671_ATP_Olambda__488,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_do(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,power_power(real,Uua),Uub)) ).

% ATP.lambda_488
tff(fact_8672_ATP_Olambda__489,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ez(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,power_power(nat,Uua),Uub)) ).

% ATP.lambda_489
tff(fact_8673_ATP_Olambda__490,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_ow(fun(nat,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,power_power(B,Uua),Uub)) ) ).

% ATP.lambda_490
tff(fact_8674_ATP_Olambda__491,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_es(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_491
tff(fact_8675_ATP_Olambda__492,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_cs(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_492
tff(fact_8676_ATP_Olambda__493,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_dc(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_493
tff(fact_8677_ATP_Olambda__494,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_gb(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_494
tff(fact_8678_ATP_Olambda__495,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_et(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,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_495
tff(fact_8679_ATP_Olambda__496,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ew(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_496
tff(fact_8680_ATP_Olambda__497,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bb(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uub),Uua) ) ) ).

% ATP.lambda_497
tff(fact_8681_ATP_Olambda__498,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_agj(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uub)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_498
tff(fact_8682_ATP_Olambda__499,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_agi(fun(A,set(B)),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uub)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_499
tff(fact_8683_ATP_Olambda__500,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_acv(fun(B,A),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_500
tff(fact_8684_ATP_Olambda__501,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_yp(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_501
tff(fact_8685_ATP_Olambda__502,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] :
      ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_tq(fun(A,real),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_502
tff(fact_8686_ATP_Olambda__503,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_tg(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_503
tff(fact_8687_ATP_Olambda__504,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_tm(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_504
tff(fact_8688_ATP_Olambda__505,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_tz(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_505
tff(fact_8689_ATP_Olambda__506,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_oi(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_506
tff(fact_8690_ATP_Olambda__507,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_ob(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_507
tff(fact_8691_ATP_Olambda__508,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_uv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_508
tff(fact_8692_ATP_Olambda__509,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_my(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,Uu,Uub),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_509
tff(fact_8693_ATP_Olambda__510,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_qf(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_510
tff(fact_8694_ATP_Olambda__511,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_up(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = divide_divide(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_511
tff(fact_8695_ATP_Olambda__512,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_si(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_512
tff(fact_8696_ATP_Olambda__513,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pj(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_513
tff(fact_8697_ATP_Olambda__514,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_uz(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_514
tff(fact_8698_ATP_Olambda__515,axiom,
    ! [Uu: fun(nat,rat),Uua: fun(nat,rat),Uub: nat] : aa(nat,rat,aa(fun(nat,rat),fun(nat,rat),aTP_Lamp_aep(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_515
tff(fact_8699_ATP_Olambda__516,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_iu(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_516
tff(fact_8700_ATP_Olambda__517,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_it(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_517
tff(fact_8701_ATP_Olambda__518,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_rq(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_518
tff(fact_8702_ATP_Olambda__519,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pm(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_519
tff(fact_8703_ATP_Olambda__520,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sh(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_520
tff(fact_8704_ATP_Olambda__521,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_jl(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_521
tff(fact_8705_ATP_Olambda__522,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_rr(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_522
tff(fact_8706_ATP_Olambda__523,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_dg(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_523
tff(fact_8707_ATP_Olambda__524,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_qy(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_524
tff(fact_8708_ATP_Olambda__525,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ps(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_525
tff(fact_8709_ATP_Olambda__526,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_fr(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_526
tff(fact_8710_ATP_Olambda__527,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_ee(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_527
tff(fact_8711_ATP_Olambda__528,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_eh(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_528
tff(fact_8712_ATP_Olambda__529,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_529
tff(fact_8713_ATP_Olambda__530,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ln(fun(B,set(A)),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_530
tff(fact_8714_ATP_Olambda__531,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ki(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_531
tff(fact_8715_ATP_Olambda__532,axiom,
    ! [A: $tType,B: $tType] :
      ( lattice(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ws(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_532
tff(fact_8716_ATP_Olambda__533,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & counta4013691401010221786attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_533
tff(fact_8717_ATP_Olambda__534,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & counta3822494911875563373attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aeu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_534
tff(fact_8718_ATP_Olambda__535,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_agf(fun(A,set(B)),fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uub)),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_535
tff(fact_8719_ATP_Olambda__536,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_536
tff(fact_8720_ATP_Olambda__537,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ads(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_537
tff(fact_8721_ATP_Olambda__538,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_wu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_538
tff(fact_8722_ATP_Olambda__539,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_539
tff(fact_8723_ATP_Olambda__540,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_hg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_540
tff(fact_8724_ATP_Olambda__541,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_nh(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_541
tff(fact_8725_ATP_Olambda__542,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_oo(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_542
tff(fact_8726_ATP_Olambda__543,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_vh(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_543
tff(fact_8727_ATP_Olambda__544,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_un(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_544
tff(fact_8728_ATP_Olambda__545,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_vi(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_545
tff(fact_8729_ATP_Olambda__546,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_qn(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_546
tff(fact_8730_ATP_Olambda__547,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_qa(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_547
tff(fact_8731_ATP_Olambda__548,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: C] : aa(C,product_prod(A,B),aa(fun(C,B),fun(C,product_prod(A,B)),aTP_Lamp_afx(fun(C,A),fun(fun(C,B),fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),product_Pair(A,B,aa(C,A,Uu,Uub)),aa(C,B,Uua,Uub)) ).

% ATP.lambda_548
tff(fact_8732_ATP_Olambda__549,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_xl(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),product_Pair(B,C,aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ).

% ATP.lambda_549
tff(fact_8733_ATP_Olambda__550,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ap(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_550
tff(fact_8734_ATP_Olambda__551,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_afm(fun(nat,set(A)),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),disjnt(A),aa(nat,set(A),Uu,Uua)),aa(nat,set(A),Uu,Uub)) ) ).

% ATP.lambda_551
tff(fact_8735_ATP_Olambda__552,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_afk(fun(A,set(B)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(set(B),$o,aa(set(B),fun(set(B),$o),disjnt(B),aa(A,set(B),Uu,Uua)),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_552
tff(fact_8736_ATP_Olambda__553,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,B)),Uua: fun(A,nat),Uub: A] : aa(A,fun(B,B),aa(fun(A,nat),fun(A,fun(B,B)),aTP_Lamp_mp(fun(A,fun(B,B)),fun(fun(A,nat),fun(A,fun(B,B))),Uu),Uua),Uub) = aa(fun(B,B),fun(B,B),aa(nat,fun(fun(B,B),fun(B,B)),compow(fun(B,B)),aa(A,nat,Uua,Uub)),aa(A,fun(B,B),Uu,Uub)) ).

% ATP.lambda_553
tff(fact_8737_ATP_Olambda__554,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aik(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,$o,Uu,Uub)
            | aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_554
tff(fact_8738_ATP_Olambda__555,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aa(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_555
tff(fact_8739_ATP_Olambda__556,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ab(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_556
tff(fact_8740_ATP_Olambda__557,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_adf(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_557
tff(fact_8741_ATP_Olambda__558,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_sq(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_558
tff(fact_8742_ATP_Olambda__559,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ss(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
      <=> aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_559
tff(fact_8743_ATP_Olambda__560,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_tc(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_560
tff(fact_8744_ATP_Olambda__561,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_adm(A,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uua,Uu) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_561
tff(fact_8745_ATP_Olambda__562,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] : aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_cg(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uua,Uub))) ) ).

% ATP.lambda_562
tff(fact_8746_ATP_Olambda__563,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uq(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_563
tff(fact_8747_ATP_Olambda__564,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_vn(fun(A,fun(A,$o)),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & ! [Y4: A] :
            ( aa(A,$o,Uua,Y4)
           => aa(A,$o,aa(A,fun(A,$o),Uu,Uub),Y4) ) ) ) ).

% ATP.lambda_564
tff(fact_8748_ATP_Olambda__565,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_pr(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_565
tff(fact_8749_ATP_Olambda__566,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( aa(nat,$o,aa(set(A),fun(nat,$o),aTP_Lamp_tr(fun(nat,set(A)),fun(set(A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(nat,set(A),Uu,Uub)),Uua) ) ) ).

% ATP.lambda_566
tff(fact_8750_ATP_Olambda__567,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_be(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_567
tff(fact_8751_ATP_Olambda__568,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_lw(fun(B,set(A)),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua) ) ).

% ATP.lambda_568
tff(fact_8752_ATP_Olambda__569,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ub(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_569
tff(fact_8753_ATP_Olambda__570,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ty(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_570
tff(fact_8754_ATP_Olambda__571,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_571
tff(fact_8755_ATP_Olambda__572,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_to(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_572
tff(fact_8756_ATP_Olambda__573,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_cd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uu,Uub),Uua) ) ).

% ATP.lambda_573
tff(fact_8757_ATP_Olambda__574,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_dj(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_574
tff(fact_8758_ATP_Olambda__575,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_pi(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = divide_divide(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_575
tff(fact_8759_ATP_Olambda__576,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_qs(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uua,Uub),Uu) ) ).

% ATP.lambda_576
tff(fact_8760_ATP_Olambda__577,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_tk(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_577
tff(fact_8761_ATP_Olambda__578,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ti(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_578
tff(fact_8762_ATP_Olambda__579,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_579
tff(fact_8763_ATP_Olambda__580,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_sm(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_580
tff(fact_8764_ATP_Olambda__581,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_pn(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_581
tff(fact_8765_ATP_Olambda__582,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_fm(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_582
tff(fact_8766_ATP_Olambda__583,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_qq(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_583
tff(fact_8767_ATP_Olambda__584,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_pa(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_584
tff(fact_8768_ATP_Olambda__585,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_mc(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),minus_minus(set(A),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_585
tff(fact_8769_ATP_Olambda__586,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_pt(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_586
tff(fact_8770_ATP_Olambda__587,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_na(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_587
tff(fact_8771_ATP_Olambda__588,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_sj(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_588
tff(fact_8772_ATP_Olambda__589,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_pz(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_589
tff(fact_8773_ATP_Olambda__590,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_sf(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_590
tff(fact_8774_ATP_Olambda__591,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_pb(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_591
tff(fact_8775_ATP_Olambda__592,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lg(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_592
tff(fact_8776_ATP_Olambda__593,axiom,
    ! [B: $tType,A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_kn(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_593
tff(fact_8777_ATP_Olambda__594,axiom,
    ! [A: $tType,B: $tType] :
      ( counta4013691401010221786attice(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_adx(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_594
tff(fact_8778_ATP_Olambda__595,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] : aa(B,set(A),aa(set(A),fun(B,set(A)),aTP_Lamp_lq(fun(B,set(A)),fun(set(A),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(B,set(A),Uu,Uub)),Uua) ).

% ATP.lambda_595
tff(fact_8779_ATP_Olambda__596,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_kt(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_596
tff(fact_8780_ATP_Olambda__597,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wx(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_597
tff(fact_8781_ATP_Olambda__598,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_ng(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_598
tff(fact_8782_ATP_Olambda__599,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rs(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_599
tff(fact_8783_ATP_Olambda__600,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,filter(B)),Uua: filter(C),Uub: A] : aa(A,filter(product_prod(B,C)),aa(filter(C),fun(A,filter(product_prod(B,C))),aTP_Lamp_xi(fun(A,filter(B)),fun(filter(C),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,aa(A,filter(B),Uu,Uub),Uua) ).

% ATP.lambda_600
tff(fact_8784_ATP_Olambda__601,axiom,
    ! [D: $tType,A: $tType,B: $tType,C: $tType,Uu: fun(D,set(product_prod(A,C))),Uua: set(product_prod(C,B)),Uub: D] : aa(D,set(product_prod(A,B)),aa(set(product_prod(C,B)),fun(D,set(product_prod(A,B))),aTP_Lamp_ahc(fun(D,set(product_prod(A,C))),fun(set(product_prod(C,B)),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,aa(D,set(product_prod(A,C)),Uu,Uub),Uua) ).

% ATP.lambda_601
tff(fact_8785_ATP_Olambda__602,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,set(product_prod(B,A))),Uua: set(B),Uub: C] : aa(C,set(A),aa(set(B),fun(C,set(A)),aTP_Lamp_ack(fun(C,set(product_prod(B,A))),fun(set(B),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,aa(C,set(product_prod(B,A)),Uu,Uub)),Uua) ).

% ATP.lambda_602
tff(fact_8786_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: int,Uub: B] : aa(B,A,aa(int,fun(B,A),aTP_Lamp_yx(fun(B,A),fun(int,fun(B,A)),Uu),Uua),Uub) = power_int(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_603
tff(fact_8787_ATP_Olambda__604,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_ze(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_604
tff(fact_8788_ATP_Olambda__605,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_zb(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_605
tff(fact_8789_ATP_Olambda__606,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_zf(fun(A,A),fun(int,fun(A,A)),Uu),Uua),Uub) = power_int(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_606
tff(fact_8790_ATP_Olambda__607,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_zd(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_607
tff(fact_8791_ATP_Olambda__608,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: int,Uub: A] : aa(A,B,aa(int,fun(A,B),aTP_Lamp_zc(fun(A,B),fun(int,fun(A,B)),Uu),Uua),Uub) = power_int(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_608
tff(fact_8792_ATP_Olambda__609,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_akl(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => (Uua) ) ) ).

% ATP.lambda_609
tff(fact_8793_ATP_Olambda__610,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_vs(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> member(B,aa(A,B,Uu,Uub),Uua) ) ).

% ATP.lambda_610
tff(fact_8794_ATP_Olambda__611,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_yn(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu) ) ).

% ATP.lambda_611
tff(fact_8795_ATP_Olambda__612,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_st(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | (Uua) ) ) ).

% ATP.lambda_612
tff(fact_8796_ATP_Olambda__613,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_akj(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & (Uua) ) ) ).

% ATP.lambda_613
tff(fact_8797_ATP_Olambda__614,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afv(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = Uua ) ) ).

% ATP.lambda_614
tff(fact_8798_ATP_Olambda__615,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ar(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_615
tff(fact_8799_ATP_Olambda__616,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ip(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uu),Uub))) ) ).

% ATP.lambda_616
tff(fact_8800_ATP_Olambda__617,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_dp(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,power_power(real,Uu),Uub)) ) ).

% ATP.lambda_617
tff(fact_8801_ATP_Olambda__618,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_uh(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),aa(nat,real,Uua,Uub)) ) ) ).

% ATP.lambda_618
tff(fact_8802_ATP_Olambda__619,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_1(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ch(fun(A,$o),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa($o,B,zero_neq_one_of_bool(B),aa(A,$o,Uu,Uub))),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_619
tff(fact_8803_ATP_Olambda__620,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akm(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ~ aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_620
tff(fact_8804_ATP_Olambda__621,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,B),fun(nat,$o),aTP_Lamp_sx(fun(nat,A),fun(fun(nat,B),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub))) ) ) ).

% ATP.lambda_621
tff(fact_8805_ATP_Olambda__622,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_sw(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua) ) ) ).

% ATP.lambda_622
tff(fact_8806_ATP_Olambda__623,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua) ) ) ).

% ATP.lambda_623
tff(fact_8807_ATP_Olambda__624,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_uo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_624
tff(fact_8808_ATP_Olambda__625,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aat(fun(A,A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( ? [X: A] :
                ( ( Uub = aa(A,A,Uu,X) )
                & aa(A,$o,Uua,X) )
            | ? [M8: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X: A] :
                    ( member(A,X,M8)
                   => aa(A,$o,Uua,X) ) ) ) ) ) ).

% ATP.lambda_625
tff(fact_8809_ATP_Olambda__626,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: fun(A,real)] :
          ( aa(fun(A,real),$o,aa(A,fun(fun(A,real),$o),aTP_Lamp_ais(set(A),fun(A,fun(fun(A,real),$o)),Uu),Uua),Uub)
        <=> ( ! [V6: A] :
                ( ( aa(A,real,Uub,V6) != zero_zero(real) )
               => member(A,V6,Uu) )
            & aa(set(A),$o,finite_finite2(A),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),Uub)))
            & ( aa(set(A),A,aa(fun(A,A),fun(set(A),A),groups7311177749621191930dd_sum(A,A),aTP_Lamp_aex(fun(A,real),fun(A,A),Uub)),aa(fun(A,$o),set(A),collect(A),aTP_Lamp_aey(fun(A,real),fun(A,$o),Uub))) = Uua ) ) ) ) ).

% ATP.lambda_626
tff(fact_8810_ATP_Olambda__627,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),$o)),Uua: list(A),Uub: list(A)] :
          ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o)),aTP_Lamp_abu(fun(list(A),fun(list(A),$o)),fun(list(A),fun(list(A),$o))),Uu),Uua),Uub)
        <=> ( ? [Y4: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),cons(A,Y4),Ys4) ) )
            | ? [X: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),cons(A,X),Xs3) )
                & ( Uub = aa(list(A),list(A),cons(A,Y4),Ys4) )
                & aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y4) )
            | ? [X: A,Y4: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),cons(A,X),Xs3) )
                & ( Uub = aa(list(A),list(A),cons(A,Y4),Ys4) )
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),X),Y4)
                & ~ aa(A,$o,aa(A,fun(A,$o),ord_less(A),Y4),X)
                & aa(list(A),$o,aa(list(A),fun(list(A),$o),Uu,Xs3),Ys4) ) ) ) ) ).

% ATP.lambda_627
tff(fact_8811_ATP_Olambda__628,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_aax(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,finite_finite2(A),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uub)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uub),Uu) ) ) ).

% ATP.lambda_628
tff(fact_8812_ATP_Olambda__629,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_aee(set(product_prod(A,A)),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,finite_finite2(A),Uua)
        & aa(set(A),$o,finite_finite2(A),Uub)
        & ( Uub != bot_bot(set(A)) )
        & ! [X: A] :
            ( member(A,X,Uua)
           => ? [Xa3: A] :
                ( member(A,Xa3,Uub)
                & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,X),Xa3),Uu) ) ) ) ) ).

% ATP.lambda_629
tff(fact_8813_ATP_Olambda__630,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_ahs(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),image(A,A,converse(A,A,Uu)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),bot_bot(set(A)))) ).

% ATP.lambda_630
tff(fact_8814_ATP_Olambda__631,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_in(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_631
tff(fact_8815_ATP_Olambda__632,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: int,Uub: A] : aa(A,A,aa(int,fun(A,A),aTP_Lamp_za(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,minus_minus(int,Uua),one_one(int))))) ) ).

% ATP.lambda_632
tff(fact_8816_ATP_Olambda__633,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dx(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_633
tff(fact_8817_ATP_Olambda__634,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_dv(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_634
tff(fact_8818_ATP_Olambda__635,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_abb(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),minus_minus(set(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_635
tff(fact_8819_ATP_Olambda__636,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_acd(fun(A,option(B)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),minus_minus(set(A),Uua),dom(A,B,Uu))) ) ).

% ATP.lambda_636
tff(fact_8820_ATP_Olambda__637,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ace(fun(A,option(B)),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uua),dom(A,B,Uu))) ) ).

% ATP.lambda_637
tff(fact_8821_ATP_Olambda__638,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_re(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = divide_divide(real,Uua,aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_638
tff(fact_8822_ATP_Olambda__639,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dw(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_639
tff(fact_8823_ATP_Olambda__640,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_ts(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu,Uua)) ) ).

% ATP.lambda_640
tff(fact_8824_ATP_Olambda__641,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [Uu: fun(A,B),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_xs(fun(A,B),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(B,Uub,aa(set(A),set(B),image2(A,B,Uu),Uua)) ) ) ).

% ATP.lambda_641
tff(fact_8825_ATP_Olambda__642,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_lu(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_642
tff(fact_8826_ATP_Olambda__643,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_lt(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_643
tff(fact_8827_ATP_Olambda__644,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_tw(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_644
tff(fact_8828_ATP_Olambda__645,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_645
tff(fact_8829_ATP_Olambda__646,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tx(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_646
tff(fact_8830_ATP_Olambda__647,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tl(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_647
tff(fact_8831_ATP_Olambda__648,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_tj(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_648
tff(fact_8832_ATP_Olambda__649,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_adr(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less(A),Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_649
tff(fact_8833_ATP_Olambda__650,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_tn(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_650
tff(fact_8834_ATP_Olambda__651,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_th(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_651
tff(fact_8835_ATP_Olambda__652,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_sn(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_652
tff(fact_8836_ATP_Olambda__653,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_cc(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_653
tff(fact_8837_ATP_Olambda__654,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_fn(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_654
tff(fact_8838_ATP_Olambda__655,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_qr(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_655
tff(fact_8839_ATP_Olambda__656,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_oz(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_656
tff(fact_8840_ATP_Olambda__657,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_po(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_657
tff(fact_8841_ATP_Olambda__658,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linordered_field(B)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_wd(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_658
tff(fact_8842_ATP_Olambda__659,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_ls(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),minus_minus(set(A),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_659
tff(fact_8843_ATP_Olambda__660,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ri(fun(A,nat),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,Uua),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_660
tff(fact_8844_ATP_Olambda__661,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lh(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_661
tff(fact_8845_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType] :
      ( counta4013691401010221786attice(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_adv(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),sup_sup(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_662
tff(fact_8846_ATP_Olambda__663,axiom,
    ! [A: $tType,B: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_kl(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_663
tff(fact_8847_ATP_Olambda__664,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_lr(set(A),fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_664
tff(fact_8848_ATP_Olambda__665,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ks(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),inf_inf(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_665
tff(fact_8849_ATP_Olambda__666,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_dk(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_666
tff(fact_8850_ATP_Olambda__667,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: filter(B),Uua: fun(A,filter(C)),Uub: A] : aa(A,filter(product_prod(B,C)),aa(fun(A,filter(C)),fun(A,filter(product_prod(B,C))),aTP_Lamp_xh(filter(B),fun(fun(A,filter(C)),fun(A,filter(product_prod(B,C)))),Uu),Uua),Uub) = prod_filter(B,C,Uu,aa(A,filter(C),Uua,Uub)) ).

% ATP.lambda_667
tff(fact_8851_ATP_Olambda__668,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_wb(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_668
tff(fact_8852_ATP_Olambda__669,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,filter(B)),Uub: C] : aa(C,filter(A),aa(fun(C,filter(B)),fun(C,filter(A)),aTP_Lamp_vy(fun(A,B),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = filtercomap(A,B,Uu,aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_669
tff(fact_8853_ATP_Olambda__670,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: set(product_prod(A,C)),Uua: fun(D,set(product_prod(C,B))),Uub: D] : aa(D,set(product_prod(A,B)),aa(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B))),aTP_Lamp_ahb(set(product_prod(A,C)),fun(fun(D,set(product_prod(C,B))),fun(D,set(product_prod(A,B)))),Uu),Uua),Uub) = relcomp(A,C,B,Uu,aa(D,set(product_prod(C,B)),Uua,Uub)) ).

% ATP.lambda_670
tff(fact_8854_ATP_Olambda__671,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_aia(fun(B,A),fun(fun(C,filter(B)),fun(C,filter(A))),Uu),Uua),Uub) = aa(filter(B),filter(A),aa(fun(B,A),fun(filter(B),filter(A)),filtermap(B,A),Uu),aa(C,filter(B),Uua,Uub)) ).

% ATP.lambda_671
tff(fact_8855_ATP_Olambda__672,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(B,A)),Uua: fun(C,set(B)),Uub: C] : aa(C,set(A),aa(fun(C,set(B)),fun(C,set(A)),aTP_Lamp_ach(set(product_prod(B,A)),fun(fun(C,set(B)),fun(C,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,Uu),aa(C,set(B),Uua,Uub)) ).

% ATP.lambda_672
tff(fact_8856_ATP_Olambda__673,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_sv($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_673
tff(fact_8857_ATP_Olambda__674,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_li(fun(B,set(A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(B,set(A),Uu,Uua)) ) ).

% ATP.lambda_674
tff(fact_8858_ATP_Olambda__675,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: fun(A,set(B)),Uub: A] : aa(A,set(B),aa(fun(A,set(B)),fun(A,set(B)),aTP_Lamp_lm(B,fun(fun(A,set(B)),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uu),aa(A,set(B),Uua,Uub)) ).

% ATP.lambda_675
tff(fact_8859_ATP_Olambda__676,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: fun(B,set(A)),Uub: B] : aa(B,set(A),aa(fun(B,set(A)),fun(B,set(A)),aTP_Lamp_le(A,fun(fun(B,set(A)),fun(B,set(A))),Uu),Uua),Uub) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),aa(B,set(A),Uua,Uub)) ).

% ATP.lambda_676
tff(fact_8860_ATP_Olambda__677,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_yr(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image2(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_677
tff(fact_8861_ATP_Olambda__678,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,set(C),aa(fun(B,C),fun(A,set(C)),aTP_Lamp_wy(fun(A,set(B)),fun(fun(B,C),fun(A,set(C))),Uu),Uua),Uub) = aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub)) ) ).

% ATP.lambda_678
tff(fact_8862_ATP_Olambda__679,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_su($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_679
tff(fact_8863_ATP_Olambda__680,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_aek(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        | aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_680
tff(fact_8864_ATP_Olambda__681,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_akk($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_681
tff(fact_8865_ATP_Olambda__682,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_zo(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_682
tff(fact_8866_ATP_Olambda__683,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_ade(fun(list(A),A),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_683
tff(fact_8867_ATP_Olambda__684,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_yu(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_684
tff(fact_8868_ATP_Olambda__685,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hr(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,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_685
tff(fact_8869_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_ul(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_686
tff(fact_8870_ATP_Olambda__687,axiom,
    ! [Uu: real,Uua: real,Uub: product_unit] : aa(product_unit,real,aa(real,fun(product_unit,real),aTP_Lamp_ajv(real,fun(real,fun(product_unit,real)),Uu),Uua),Uub) = powr_real(Uu,Uua) ).

% ATP.lambda_687
tff(fact_8871_ATP_Olambda__688,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,set(A),aa(A,fun(A,set(A)),aTP_Lamp_akh(set(product_prod(A,A)),fun(A,fun(A,set(A))),Uu),Uua),Uub) = order_underS(A,Uu,Uua) ).

% ATP.lambda_688
tff(fact_8872_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: set(B),Uub: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aTP_Lamp_agd(B,fun(set(B),fun(A,set(B))),Uu),Uua),Uub) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uu),Uua) ).

% ATP.lambda_689
tff(fact_8873_ATP_Olambda__690,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] : aa(A,set(A),aa(set(B),fun(A,set(A)),aTP_Lamp_agk(fun(B,A),fun(set(B),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image2(B,A,Uu),Uua) ).

% ATP.lambda_690
tff(fact_8874_ATP_Olambda__691,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_qx(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_691
tff(fact_8875_ATP_Olambda__692,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_py(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_692
tff(fact_8876_ATP_Olambda__693,axiom,
    ! [Uu: fun(nat,real),Uua: nat,Uub: nat] : aa(nat,real,aa(nat,fun(nat,real),aTP_Lamp_ahz(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_693
tff(fact_8877_ATP_Olambda__694,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_sl(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_694
tff(fact_8878_ATP_Olambda__695,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dn(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_695
tff(fact_8879_ATP_Olambda__696,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tu(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_696
tff(fact_8880_ATP_Olambda__697,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_qk(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_697
tff(fact_8881_ATP_Olambda__698,axiom,
    ! [C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: B,Uua: fun(B,C),Uub: B] : aa(B,C,aa(fun(B,C),fun(B,C),aTP_Lamp_qc(B,fun(fun(B,C),fun(B,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),Uub)) ) ).

% ATP.lambda_698
tff(fact_8882_ATP_Olambda__699,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_pc(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_699
tff(fact_8883_ATP_Olambda__700,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_fp(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_700
tff(fact_8884_ATP_Olambda__701,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,nat),Uub: nat] : aa(nat,real,aa(fun(nat,nat),fun(nat,real),aTP_Lamp_yg(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),Uu),Uua),Uub) = aa(nat,real,Uu,aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_701
tff(fact_8885_ATP_Olambda__702,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_yj(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_702
tff(fact_8886_ATP_Olambda__703,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_md(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_703
tff(fact_8887_ATP_Olambda__704,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,$o),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_akx(fun(B,$o),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_704
tff(fact_8888_ATP_Olambda__705,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_akr(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ).

% ATP.lambda_705
tff(fact_8889_ATP_Olambda__706,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_afe(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_706
tff(fact_8890_ATP_Olambda__707,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_ka(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_707
tff(fact_8891_ATP_Olambda__708,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_ady(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_8892_ATP_Olambda__709,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( condit1219197933456340205attice(A)
        & condit1219197933456340205attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xv(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_709
tff(fact_8893_ATP_Olambda__710,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_wn(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_710
tff(fact_8894_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_wq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_711
tff(fact_8895_ATP_Olambda__712,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_nz(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_712
tff(fact_8896_ATP_Olambda__713,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_rp(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_713
tff(fact_8897_ATP_Olambda__714,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_uw(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_714
tff(fact_8898_ATP_Olambda__715,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aet(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_715
tff(fact_8899_ATP_Olambda__716,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_xr(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_716
tff(fact_8900_ATP_Olambda__717,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_sr(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_717
tff(fact_8901_ATP_Olambda__718,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pg(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_718
tff(fact_8902_ATP_Olambda__719,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_yh(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_719
tff(fact_8903_ATP_Olambda__720,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_yi(fun(nat,nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,Uu,Uub)) ) ).

% ATP.lambda_720
tff(fact_8904_ATP_Olambda__721,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,$o),Uub: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aTP_Lamp_aie(fun(B,A),fun(fun(A,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uua,aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_721
tff(fact_8905_ATP_Olambda__722,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_mg(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_722
tff(fact_8906_ATP_Olambda__723,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_px(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_723
tff(fact_8907_ATP_Olambda__724,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_sa(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_724
tff(fact_8908_ATP_Olambda__725,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_sp(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_725
tff(fact_8909_ATP_Olambda__726,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_abs(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_726
tff(fact_8910_ATP_Olambda__727,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_abr(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_727
tff(fact_8911_ATP_Olambda__728,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_la(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_728
tff(fact_8912_ATP_Olambda__729,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vr(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ).

% ATP.lambda_729
tff(fact_8913_ATP_Olambda__730,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_xm(fun(B,$o),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uu,Uua) ) ).

% ATP.lambda_730
tff(fact_8914_ATP_Olambda__731,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] : aa(A,fun(product_prod(A,A),$o),aa(A,fun(A,fun(product_prod(A,A),$o)),aTP_Lamp_aed(set(product_prod(A,A)),fun(A,fun(A,fun(product_prod(A,A),$o))),Uu),Uua),Uub) = aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_aec(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)) ).

% ATP.lambda_731
tff(fact_8915_ATP_Olambda__732,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_oy(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_ox(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_732
tff(fact_8916_ATP_Olambda__733,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_ov(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_ou(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_733
tff(fact_8917_ATP_Olambda__734,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uua: A,Uub: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_akz(fun(A,fun(A,A)),fun(A,fun(option(A),option(A))),Uu),Uua),Uub) = aa(A,option(A),some(A),case_option(A,A,Uua,aa(A,fun(A,A),Uu,Uua),Uub)) ).

% ATP.lambda_734
tff(fact_8918_ATP_Olambda__735,axiom,
    ! [B: $tType] :
      ( real_V4867850818363320053vector(B)
     => ! [Uu: set(B),Uua: B,Uub: B] :
          ( aa(B,$o,aa(B,fun(B,$o),aTP_Lamp_aiy(set(B),fun(B,fun(B,$o)),Uu),Uua),Uub)
        <=> ( aa(B,real,real_V7696804695334737415tation(B,real_V4986007116245087402_basis(B,Uu),Uua),Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_735
tff(fact_8919_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aiv(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,real,real_V7696804695334737415tation(A,Uu,Uua),Uub) != zero_zero(real) ) ) ) ).

% ATP.lambda_736
tff(fact_8920_ATP_Olambda__737,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_wf(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),Uub)),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_737
tff(fact_8921_ATP_Olambda__738,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_we(A,fun(set(A),fun(A,filter(A))),Uu),Uua),Uub) = principal(A,aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),Uub)),Uua)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_738
tff(fact_8922_ATP_Olambda__739,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_vo(A,fun(set(A),fun(set(A),filter(A))),Uu),Uua),Uub) = principal(A,aa(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),aa(A,fun(set(A),set(A)),insert2(A),Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_739
tff(fact_8923_ATP_Olambda__740,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_ct(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,power_power(A,Uua),Uub))) ) ).

% ATP.lambda_740
tff(fact_8924_ATP_Olambda__741,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_ho(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_741
tff(fact_8925_ATP_Olambda__742,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_mn(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) != Uua ) ) ).

% ATP.lambda_742
tff(fact_8926_ATP_Olambda__743,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_mo(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uua,Uub) != Uu ) ) ).

% ATP.lambda_743
tff(fact_8927_ATP_Olambda__744,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_xa(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Sup_Sup(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_744
tff(fact_8928_ATP_Olambda__745,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( condit1219197933456340205attice(C)
     => ! [Uu: fun(A,set(B)),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_wz(fun(A,set(B)),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,Uua),aa(A,set(B),Uu,Uub))) ) ).

% ATP.lambda_745
tff(fact_8929_ATP_Olambda__746,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_dl(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ~ aa(B,$o,aa(B,fun(B,$o),dvd_dvd(B),Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_746
tff(fact_8930_ATP_Olambda__747,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A] : aa(A,nat,aa(fun(A,fun(B,$o)),fun(A,nat),aTP_Lamp_eg(set(B),fun(fun(A,fun(B,$o)),fun(A,nat)),Uu),Uua),Uub) = aa(set(B),nat,finite_card(B),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_cm(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub))) ).

% ATP.lambda_747
tff(fact_8931_ATP_Olambda__748,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(list(B),fun(product_prod(A,B),$o),aTP_Lamp_abz(list(A),fun(list(B),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [I4: nat] :
          ( ( Uub = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Uu),I4)),aa(nat,B,nth(B,Uua),I4)) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(list(B),nat,size_size(list(B)),Uua))) ) ) ).

% ATP.lambda_748
tff(fact_8932_ATP_Olambda__749,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( aa(A,$o,aa(set(nat),fun(A,$o),aTP_Lamp_aav(list(A),fun(set(nat),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [I4: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I4) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Uu))
          & member(nat,I4,Uua) ) ) ).

% ATP.lambda_749
tff(fact_8933_ATP_Olambda__750,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aaf(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A9: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A9) )
              & member(A,A9,Uu) ) ) ) ).

% ATP.lambda_750
tff(fact_8934_ATP_Olambda__751,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aad(set(A),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A9: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A9) )
              & member(A,A9,Uu) ) ) ) ).

% ATP.lambda_751
tff(fact_8935_ATP_Olambda__752,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ain(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [K3: real] : member(A,aa(A,A,minus_minus(A,Uub),aa(A,A,real_V8093663219630862766scaleR(A,K3),Uu)),real_Vector_span(A,Uua)) ) ) ).

% ATP.lambda_752
tff(fact_8936_ATP_Olambda__753,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_zs(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( ( Uub = aa(B,A,Uu,X) )
          & member(B,X,Uua) ) ) ).

% ATP.lambda_753
tff(fact_8937_ATP_Olambda__754,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_zt(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( ( Uub = aa(B,A,Uu,X) )
          & aa(B,$o,Uua,X) ) ) ).

% ATP.lambda_754
tff(fact_8938_ATP_Olambda__755,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,B),Uub: B] :
      ( aa(B,$o,aa(fun(A,B),fun(B,$o),aTP_Lamp_zp(fun(A,$o),fun(fun(A,B),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X: A] :
          ( ( Uub = aa(A,B,Uua,X) )
          & aa(A,$o,Uu,X) ) ) ).

% ATP.lambda_755
tff(fact_8939_ATP_Olambda__756,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zz(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X: A] :
          ( member(A,X,Uu)
         => aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X) ) ) ).

% ATP.lambda_756
tff(fact_8940_ATP_Olambda__757,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(A,fun(B,$o)),fun(B,$o),aTP_Lamp_akw(set(A),fun(fun(A,fun(B,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X: A] :
          ( member(A,X,Uu)
         => aa(B,$o,aa(A,fun(B,$o),Uua,X),Uub) ) ) ).

% ATP.lambda_757
tff(fact_8941_ATP_Olambda__758,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aef(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X: A] :
          ( member(A,X,Uu)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X) ) ) ).

% ATP.lambda_758
tff(fact_8942_ATP_Olambda__759,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(A,fun(B,$o)),fun(B,$o),aTP_Lamp_aky(set(A),fun(fun(A,fun(B,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X: A] :
          ( member(A,X,Uu)
          & aa(B,$o,aa(A,fun(B,$o),Uua,X),Uub) ) ) ).

% ATP.lambda_759
tff(fact_8943_ATP_Olambda__760,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aeg(fun(B,fun(A,$o)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( member(B,X,Uua)
          & aa(A,$o,aa(B,fun(A,$o),Uu,X),Uub) ) ) ).

% ATP.lambda_760
tff(fact_8944_ATP_Olambda__761,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ahu(set(product_prod(A,A)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ! [X: product_prod(A,A)] :
          ( member(product_prod(A,A),X,Uu)
         => aa(product_prod(A,A),$o,aa(fun(A,fun(A,$o)),fun(product_prod(A,A),$o),product_case_prod(A,A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_aht(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub)),X) ) ) ).

% ATP.lambda_761
tff(fact_8945_ATP_Olambda__762,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aei(set(product_prod(B,A)),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( member(B,X,Uua)
          & member(product_prod(B,A),aa(A,product_prod(B,A),product_Pair(B,A,X),Uub),Uu) ) ) ).

% ATP.lambda_762
tff(fact_8946_ATP_Olambda__763,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aft(fun(A,B),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: A] :
          ( member(A,X,Uua)
          & ( aa(A,B,Uu,X) = aa(A,B,Uu,Uub) ) ) ) ).

% ATP.lambda_763
tff(fact_8947_ATP_Olambda__764,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_us(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [N: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),N)
             => aa(real,$o,aa(real,fun(real,$o),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,N)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_764
tff(fact_8948_ATP_Olambda__765,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aej(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( member(B,X,Uua)
          & ( Uub = aa(B,A,Uu,X) ) ) ) ).

% ATP.lambda_765
tff(fact_8949_ATP_Olambda__766,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_vl(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [A9: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uub),A9)
             => ! [B7: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),A9),B7)
                 => aa(real,$o,aa(real,fun(real,$o),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,A9,B7)))),aa(nat,real,Uua,A9)) ) ) ) ) ).

% ATP.lambda_766
tff(fact_8950_ATP_Olambda__767,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: filter(B),Uub: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aTP_Lamp_abn(fun(A,B),fun(filter(B),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ? [Q7: fun(B,$o)] :
          ( aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Q7),Uua)
          & ! [X: A] :
              ( aa(B,$o,Q7,aa(A,B,Uu,X))
             => aa(A,$o,Uub,X) ) ) ) ).

% ATP.lambda_767
tff(fact_8951_ATP_Olambda__768,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_zk(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [Y4: A] :
          ( aa(A,$o,Uu,Y4)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),Y4) ) ) ).

% ATP.lambda_768
tff(fact_8952_ATP_Olambda__769,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aaj(fun(A,A),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ? [N: 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)),N),Uu),Uua) ) ).

% ATP.lambda_769
tff(fact_8953_ATP_Olambda__770,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aim(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [X: A,Y4: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y4) )
              & member(A,X,real_Vector_span(A,Uu))
              & member(A,Y4,real_Vector_span(A,Uua)) ) ) ) ).

% ATP.lambda_770
tff(fact_8954_ATP_Olambda__771,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aag(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A9: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A9),B7) )
              & member(A,A9,Uu)
              & member(A,B7,Uua) ) ) ) ).

% ATP.lambda_771
tff(fact_8955_ATP_Olambda__772,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aae(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ? [A9: A,B7: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A9),B7) )
              & member(A,A9,Uu)
              & member(A,B7,Uua) ) ) ) ).

% ATP.lambda_772
tff(fact_8956_ATP_Olambda__773,axiom,
    ! [A: $tType,Uu: filter(A),Uua: filter(A),Uub: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(A),fun(fun(A,$o),$o),aTP_Lamp_abo(filter(A),fun(filter(A),fun(fun(A,$o),$o)),Uu),Uua),Uub)
    <=> ? [Q7: fun(A,$o),R6: fun(A,$o)] :
          ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Q7),Uu)
          & aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),R6),Uua)
          & ! [X: A] :
              ( ( aa(A,$o,Q7,X)
                & aa(A,$o,R6,X) )
             => aa(A,$o,Uub,X) ) ) ) ).

% ATP.lambda_773
tff(fact_8957_ATP_Olambda__774,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_fx(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc)))),semiring_char_0_fact(real,Uub))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_774
tff(fact_8958_ATP_Olambda__775,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_ft(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & ~ aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),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),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,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc)))),semiring_char_0_fact(real,Uub)))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_775
tff(fact_8959_ATP_Olambda__776,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_fv(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),dvd_dvd(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),binomial(Uub,Uuc)))),semiring_char_0_fact(real,Uub))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_776
tff(fact_8960_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: product_prod(C,A),Uua: A,Uub: B,Uuc: set(product_prod(C,B))] :
      aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_ahe(product_prod(C,A),fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = $ite(aa(product_prod(C,A),A,product_snd(C,A),Uu) = Uua,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert2(product_prod(C,B)),aa(B,product_prod(C,B),product_Pair(C,B,aa(product_prod(C,A),C,product_fst(C,A),Uu)),Uub)),Uuc),Uuc) ).

% ATP.lambda_777
tff(fact_8961_ATP_Olambda__778,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_yt(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(member(B,Uuc,aa(set(A),set(B),image2(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_778
tff(fact_8962_ATP_Olambda__779,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: A,Uub: B,Uuc: set(B)] :
      aa(set(B),set(B),aa(B,fun(set(B),set(B)),aa(A,fun(B,fun(set(B),set(B))),aTP_Lamp_agp(set(A),fun(A,fun(B,fun(set(B),set(B)))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uua,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),Uuc),Uuc) ).

% ATP.lambda_779
tff(fact_8963_ATP_Olambda__780,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_fc(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_780
tff(fact_8964_ATP_Olambda__781,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_hs(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_781
tff(fact_8965_ATP_Olambda__782,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_wm(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_782
tff(fact_8966_ATP_Olambda__783,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_ht(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_783
tff(fact_8967_ATP_Olambda__784,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_fd(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_784
tff(fact_8968_ATP_Olambda__785,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(set(A),fun(fun(A,B),fun(A,B)),aTP_Lamp_yq(fun(A,B),fun(set(A),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uua),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_785
tff(fact_8969_ATP_Olambda__786,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_fs(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_786
tff(fact_8970_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_hn(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_787
tff(fact_8971_ATP_Olambda__788,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_db(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_788
tff(fact_8972_ATP_Olambda__789,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: B,Uuc: A] :
          aa(A,B,aa(B,fun(A,B),aa(fun(A,B),fun(B,fun(A,B)),aTP_Lamp_hp(A,fun(fun(A,B),fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),Uub) ) ).

% ATP.lambda_789
tff(fact_8973_ATP_Olambda__790,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: B,Uub: B,Uuc: A] :
      aa(A,B,aa(B,fun(A,B),aa(B,fun(B,fun(A,B)),aTP_Lamp_afs(set(A),fun(B,fun(B,fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(member(A,Uuc,Uu),Uua,Uub) ).

% ATP.lambda_790
tff(fact_8974_ATP_Olambda__791,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
      aa(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_jw(fun(B,$o),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = $ite(aa(B,$o,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ).

% ATP.lambda_791
tff(fact_8975_ATP_Olambda__792,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,option(B)),Uub: fun(A,option(B)),Uuc: A] :
      aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),aa(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B))),aTP_Lamp_acb(fun(A,$o),fun(fun(A,option(B)),fun(fun(A,option(B)),fun(A,option(B)))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,option(B),Uua,Uuc),aa(A,option(B),Uub,Uuc)) ).

% ATP.lambda_792
tff(fact_8976_ATP_Olambda__793,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_hl(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_793
tff(fact_8977_ATP_Olambda__794,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,$o),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_cy(fun(A,$o),fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uu,Uuc),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_794
tff(fact_8978_ATP_Olambda__795,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
      aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_vv(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ).

% ATP.lambda_795
tff(fact_8979_ATP_Olambda__796,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: fun(A,$o),Uuc: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aa(fun(A,B),fun(fun(A,$o),fun(A,B)),aTP_Lamp_yo(fun(A,B),fun(fun(A,B),fun(fun(A,$o),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uub,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uua,Uuc)) ) ).

% ATP.lambda_796
tff(fact_8980_ATP_Olambda__797,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,C)),Uua: A,Uub: B,Uuc: set(product_prod(A,C))] : aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(B,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aTP_Lamp_agy(set(product_prod(B,C)),fun(A,fun(B,fun(set(product_prod(A,C)),set(product_prod(A,C))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(B,C),set(product_prod(A,C)),aa(fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),fun(product_prod(B,C),fun(set(product_prod(A,C)),set(product_prod(A,C)))),product_case_prod(B,C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_agx(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_797
tff(fact_8981_ATP_Olambda__798,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: C,Uub: A,Uuc: set(product_prod(C,B))] : aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(A,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aTP_Lamp_agr(set(product_prod(A,B)),fun(C,fun(A,fun(set(product_prod(C,B)),set(product_prod(C,B))))),Uu),Uua),Uub),Uuc) = finite_fold(product_prod(A,B),set(product_prod(C,B)),aa(fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),fun(product_prod(A,B),fun(set(product_prod(C,B)),set(product_prod(C,B)))),product_case_prod(A,B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_agq(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uua),Uub)),Uuc,Uu) ).

% ATP.lambda_798
tff(fact_8982_ATP_Olambda__799,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_je(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_799
tff(fact_8983_ATP_Olambda__800,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: fun(D,B),Uuc: D] : aa(D,C,aa(fun(D,B),fun(D,C),aa(fun(D,A),fun(fun(D,B),fun(D,C)),aTP_Lamp_ajk(fun(A,fun(B,C)),fun(fun(D,A),fun(fun(D,B),fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),aa(D,B,Uub,Uuc)) ) ).

% ATP.lambda_800
tff(fact_8984_ATP_Olambda__801,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,A),Uub: B,Uuc: D] : aa(D,C,aa(B,fun(D,C),aa(fun(D,A),fun(B,fun(D,C)),aTP_Lamp_ajj(fun(A,fun(B,C)),fun(fun(D,A),fun(B,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,aa(D,A,Uua,Uuc)),Uub) ) ).

% ATP.lambda_801
tff(fact_8985_ATP_Olambda__802,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aji(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_802
tff(fact_8986_ATP_Olambda__803,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [Uu: fun(A,fun(B,C)),Uua: fun(D,B),Uub: A,Uuc: D] : aa(D,C,aa(A,fun(D,C),aa(fun(D,B),fun(A,fun(D,C)),aTP_Lamp_aan(fun(A,fun(B,C)),fun(fun(D,B),fun(A,fun(D,C))),Uu),Uua),Uub),Uuc) = aa(B,C,aa(A,fun(B,C),Uu,Uub),aa(D,B,Uua,Uuc)) ) ).

% ATP.lambda_803
tff(fact_8987_ATP_Olambda__804,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_ed(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_ec(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_804
tff(fact_8988_ATP_Olambda__805,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(B,A),Uub: filter(B),Uuc: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(filter(B),fun(fun(A,$o),$o),aa(fun(B,A),fun(filter(B),fun(fun(A,$o),$o)),aTP_Lamp_abm(set(B),fun(fun(B,A),fun(filter(B),fun(fun(A,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),aa(fun(A,$o),fun(B,$o),aa(fun(B,A),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_abl(set(B),fun(fun(B,A),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_805
tff(fact_8989_ATP_Olambda__806,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_fe(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(
                Uuc = zero_zero(nat),
                aa(A,A,uminus_uminus(A),Uub),
                $ite(Uuc = Uu,one_one(A),zero_zero(A)) )),
            aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_806
tff(fact_8990_ATP_Olambda__807,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V4867850818363320053vector(A)
        & real_V4867850818363320053vector(B) )
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B,Uuc: B] :
          aa(B,A,aa(B,fun(B,A),aa(fun(B,A),fun(B,fun(B,A)),aTP_Lamp_aix(set(B),fun(fun(B,A),fun(B,fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,real_V8093663219630862766scaleR(A,aa(B,real,real_V7696804695334737415tation(B,real_V4986007116245087402_basis(B,Uu),Uub),Uuc)),
            $ite(member(B,Uuc,Uu),aa(B,A,Uua,Uuc),zero_zero(A))) ) ).

% ATP.lambda_807
tff(fact_8991_ATP_Olambda__808,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,fun(A,$o)),Uub: set(A),Uuc: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aa(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o)),aTP_Lamp_akc(set(A),fun(fun(A,fun(A,$o)),fun(set(A),fun(set(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(A),$o,pred_chain(A,Uu,Uua),Uuc)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less(set(A)),Uub),Uuc) ) ) ).

% ATP.lambda_808
tff(fact_8992_ATP_Olambda__809,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_ha(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_gz(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ax(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_809
tff(fact_8993_ATP_Olambda__810,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_cp(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aTP_Lamp_co(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ax(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_810
tff(fact_8994_ATP_Olambda__811,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_fb(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_fa(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,nat,power_power(nat,Uub),Uuc)) ).

% ATP.lambda_811
tff(fact_8995_ATP_Olambda__812,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_ey(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_ex(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_812
tff(fact_8996_ATP_Olambda__813,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_np(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),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,power_power(real,Uub),Uuc)) ).

% ATP.lambda_813
tff(fact_8997_ATP_Olambda__814,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_nn(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),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,power_power(real,Uub),Uuc)) ).

% ATP.lambda_814
tff(fact_8998_ATP_Olambda__815,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_nm(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),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uua),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uub),Uua)),Uuc)) ).

% ATP.lambda_815
tff(fact_8999_ATP_Olambda__816,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_nl(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),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uub),semiring_char_0_fact(real,Uuc))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uua),Uub)),Uuc)) ).

% ATP.lambda_816
tff(fact_9000_ATP_Olambda__817,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_ec(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,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_817
tff(fact_9001_ATP_Olambda__818,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_rj(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = divide_divide(A,aa(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_818
tff(fact_9002_ATP_Olambda__819,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(set(product_prod(A,A)),fun(A,fun(A,$o)),aTP_Lamp_ja(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc))
          & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Uub),Uuc),Uua) ) ) ) ).

% ATP.lambda_819
tff(fact_9003_ATP_Olambda__820,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_ajh(set(product_prod(B,B)),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> member(product_prod(B,B),aa(B,product_prod(B,B),product_Pair(B,B,aa(A,B,Uua,Uub)),aa(A,B,Uua,Uuc)),Uu) ) ).

% ATP.lambda_820
tff(fact_9004_ATP_Olambda__821,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_td(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_821
tff(fact_9005_ATP_Olambda__822,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(B,fun(real,fun(A,$o)),aTP_Lamp_ug(fun(A,B),fun(B,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub) ) ) ).

% ATP.lambda_822
tff(fact_9006_ATP_Olambda__823,axiom,
    ! [A: $tType,B: $tType] :
      ( order(A)
     => ! [Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] : aa(A,set(B),aa(set(B),fun(A,set(B)),aa(set(B),fun(set(B),fun(A,set(B))),aTP_Lamp_yk(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(A,set(B),Uu,Uuc)),Uua)),Uub) ) ).

% ATP.lambda_823
tff(fact_9007_ATP_Olambda__824,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(set(A),set(A)),Uub: set(A),Uuc: set(A)] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),aa(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A))),aTP_Lamp_ajm(set(A),fun(fun(set(A),set(A)),fun(set(A),fun(set(A),set(A)))),Uu),Uua),Uub),Uuc) = 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),Uua,Uuc)),Uub)),Uu) ).

% ATP.lambda_824
tff(fact_9008_ATP_Olambda__825,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_kb(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,aa(set(B),set(A),image2(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_825
tff(fact_9009_ATP_Olambda__826,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_cm(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_826
tff(fact_9010_ATP_Olambda__827,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ax(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_827
tff(fact_9011_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: fun(A,real),Uuc: A] :
          ( aa(A,$o,aa(fun(A,real),fun(A,$o),aa(fun(A,real),fun(fun(A,real),fun(A,$o)),aTP_Lamp_va(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & aa(real,$o,aa(real,fun(real,$o),ord_less(real),aa(A,real,Uua,Uuc)),aa(A,real,Uub,Uuc)) ) ) ) ).

% ATP.lambda_828
tff(fact_9012_ATP_Olambda__829,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_jz(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_829
tff(fact_9013_ATP_Olambda__830,axiom,
    ! [A: $tType,C: $tType,Uu: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_kg(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_830
tff(fact_9014_ATP_Olambda__831,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_kq(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_831
tff(fact_9015_ATP_Olambda__832,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_afr(fun(A,B),fun(set(A),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uua)
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_832
tff(fact_9016_ATP_Olambda__833,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_ym(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uub)
        & member(A,aa(B,A,Uua,Uuc),Uu) ) ) ).

% ATP.lambda_833
tff(fact_9017_ATP_Olambda__834,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_aez(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_834
tff(fact_9018_ATP_Olambda__835,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_aco(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_835
tff(fact_9019_ATP_Olambda__836,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_afb(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_836
tff(fact_9020_ATP_Olambda__837,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aTP_Lamp_acs(nat,fun(nat,fun(nat,fun(nat,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uuc)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_837
tff(fact_9021_ATP_Olambda__838,axiom,
    ! [A: $tType,B: $tType,Uu: filter(A),Uua: filter(B),Uub: fun(A,$o),Uuc: fun(B,$o)] :
      ( aa(fun(B,$o),$o,aa(fun(A,$o),fun(fun(B,$o),$o),aa(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o)),aTP_Lamp_zi(filter(A),fun(filter(B),fun(fun(A,$o),fun(fun(B,$o),$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(filter(A),$o,aa(fun(A,$o),fun(filter(A),$o),eventually(A),Uub),Uu)
        & aa(filter(B),$o,aa(fun(B,$o),fun(filter(B),$o),eventually(B),Uuc),Uua) ) ) ).

% ATP.lambda_838
tff(fact_9022_ATP_Olambda__839,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_kc(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_839
tff(fact_9023_ATP_Olambda__840,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,set(product_prod(A,B))),Uua: C,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(C,fun(A,fun(B,$o)),aTP_Lamp_lj(fun(C,set(product_prod(A,B))),fun(C,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),product_Pair(A,B,Uub),Uuc),aa(C,set(product_prod(A,B)),Uu,Uua)) ) ).

% ATP.lambda_840
tff(fact_9024_ATP_Olambda__841,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_bv(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_841
tff(fact_9025_ATP_Olambda__842,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_ea(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_842
tff(fact_9026_ATP_Olambda__843,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_rn(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,divide_divide(real,one_one(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(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,minus_minus(A,Uuc),Uub))))) ) ).

% ATP.lambda_843
tff(fact_9027_ATP_Olambda__844,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_is(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,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_844
tff(fact_9028_ATP_Olambda__845,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,$o),Uub: fun(A,B),Uuc: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(B,$o),fun(fun(A,B),fun(A,$o)),aTP_Lamp_yd(set(A),fun(fun(B,$o),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(B,$o,Uua,aa(A,B,Uub,Uuc))
        & member(A,Uuc,Uu) ) ) ).

% ATP.lambda_845
tff(fact_9029_ATP_Olambda__846,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(B,A),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(fun(B,A),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_abl(set(B),fun(fun(B,A),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uub,aa(B,A,Uua,Uuc))
        & member(B,Uuc,Uu) ) ) ).

% ATP.lambda_846
tff(fact_9030_ATP_Olambda__847,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_ou(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,minus_minus(A,divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(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,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_847
tff(fact_9031_ATP_Olambda__848,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A,Uuc: nat] : aa(nat,B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_ox(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uua,Uuc)),aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_848
tff(fact_9032_ATP_Olambda__849,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_or(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)),divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu,Uua))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_849
tff(fact_9033_ATP_Olambda__850,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_fj(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,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_850
tff(fact_9034_ATP_Olambda__851,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_uk(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> member(B,aa(A,B,Uu,Uuc),aa(set(B),set(B),minus_minus(set(B),Uub),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_851
tff(fact_9035_ATP_Olambda__852,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,A),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,A),fun(A,fun(A,$o)),aTP_Lamp_aeh(fun(A,$o),fun(fun(A,A),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uuc)
        & ( Uub = aa(A,A,Uua,Uuc) ) ) ) ).

% ATP.lambda_852
tff(fact_9036_ATP_Olambda__853,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_adb(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_853
tff(fact_9037_ATP_Olambda__854,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_fa(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,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_854
tff(fact_9038_ATP_Olambda__855,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_ex(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,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_855
tff(fact_9039_ATP_Olambda__856,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: B,Uuc: C] : aa(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_lo(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(B,set(A),Uu,Uub)),aa(C,set(A),Uua,Uuc)) ).

% ATP.lambda_856
tff(fact_9040_ATP_Olambda__857,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_kj(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_857
tff(fact_9041_ATP_Olambda__858,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( counta4013691401010221786attice(C)
     => ! [Uu: fun(A,C),Uua: fun(B,C),Uub: A,Uuc: B] : aa(B,C,aa(A,fun(B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_adt(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),sup_sup(C),aa(A,C,Uu,Uub)),aa(B,C,Uua,Uuc)) ) ).

% ATP.lambda_858
tff(fact_9042_ATP_Olambda__859,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: fun(A,filter(C)),Uua: fun(B,filter(D)),Uub: A,Uuc: B] : aa(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_xj(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = prod_filter(C,D,aa(A,filter(C),Uu,Uub),aa(B,filter(D),Uua,Uuc)) ).

% ATP.lambda_859
tff(fact_9043_ATP_Olambda__860,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,$o),fun(A,fun(B,$o)),aTP_Lamp_zg(fun(A,$o),fun(fun(B,$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(B,$o,Uua,Uuc) ) ) ).

% ATP.lambda_860
tff(fact_9044_ATP_Olambda__861,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( aa(B,$o,aa(list(B),fun(B,$o),aa(fun(list(B),A),fun(list(B),fun(B,$o)),aTP_Lamp_adg(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_861
tff(fact_9045_ATP_Olambda__862,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_gy(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_cm(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_862
tff(fact_9046_ATP_Olambda__863,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_cn(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7311177749621191930dd_sum(B,C),aa(A,fun(B,C),Uua,Uuc)),aa(fun(B,$o),set(B),collect(B),aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_cm(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_863
tff(fact_9047_ATP_Olambda__864,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_oh(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_864
tff(fact_9048_ATP_Olambda__865,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_nu(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,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_865
tff(fact_9049_ATP_Olambda__866,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_nw(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,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_866
tff(fact_9050_ATP_Olambda__867,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_ro(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = divide_divide(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(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_867
tff(fact_9051_ATP_Olambda__868,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,$o)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( aa(B,$o,aa(B,fun(B,$o),aa(fun(B,A),fun(B,fun(B,$o)),aTP_Lamp_afl(fun(A,fun(A,$o)),fun(fun(B,A),fun(B,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(B,A,Uua,Uub) != aa(B,A,Uua,Uuc) )
       => aa(A,$o,aa(A,fun(A,$o),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc)) ) ) ).

% ATP.lambda_868
tff(fact_9052_ATP_Olambda__869,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aa(set(B),fun(set(B),fun(A,$o)),aTP_Lamp_aba(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( aa(set(B),$o,finite_finite2(B),aa(A,set(B),Uu,Uuc))
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc))
        & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua) ) ) ).

% ATP.lambda_869
tff(fact_9053_ATP_Olambda__870,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_uj(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub)) ) ) ).

% ATP.lambda_870
tff(fact_9054_ATP_Olambda__871,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_rx(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,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_rw(A,A)))))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_rw(A,A))))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_rw(A,A)))))) ) ).

% ATP.lambda_871
tff(fact_9055_ATP_Olambda__872,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_rt(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,minus_minus(A,Uuc),Uua)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,minus_minus(A,Uuc),Uua)))) ) ).

% ATP.lambda_872
tff(fact_9056_ATP_Olambda__873,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_ru(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,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub)))) ) ).

% ATP.lambda_873
tff(fact_9057_ATP_Olambda__874,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(B,C),Uuc: B] : aa(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_lb(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(nat,C,semiring_1_of_nat(C),aa(set(A),nat,finite_card(A),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_kq(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_874
tff(fact_9058_ATP_Olambda__875,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: fun(C,B),Uub: set(C),Uuc: A] : aa(A,set(B),aa(set(C),fun(A,set(B)),aa(fun(C,B),fun(set(C),fun(A,set(B))),aTP_Lamp_afy(fun(C,A),fun(fun(C,B),fun(set(C),fun(A,set(B)))),Uu),Uua),Uub),Uuc) = aa(set(C),set(B),image2(C,B,Uua),aa(set(C),set(C),aa(set(C),fun(set(C),set(C)),inf_inf(set(C)),vimage(C,A,Uu,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),bot_bot(set(A))))),Uub)) ).

% ATP.lambda_875
tff(fact_9059_ATP_Olambda__876,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ko(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_kg(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_876
tff(fact_9060_ATP_Olambda__877,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ku(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_kq(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_877
tff(fact_9061_ATP_Olambda__878,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_kh(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7311177749621191930dd_sum(A,B),Uua),aa(fun(A,$o),set(A),collect(A),aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_kg(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_878
tff(fact_9062_ATP_Olambda__879,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_kr(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),Uub),aa(fun(A,$o),set(A),collect(A),aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_kq(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_879
tff(fact_9063_ATP_Olambda__880,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_sk(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7311177749621191930dd_sum(nat,A),aa(A,fun(nat,A),aTP_Lamp_cs(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uua)))) ) ) ).

% ATP.lambda_880
tff(fact_9064_ATP_Olambda__881,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( counta4013691401010221786attice(C)
     => ! [Uu: set(B),Uua: fun(A,C),Uub: fun(B,C),Uuc: A] : aa(A,C,aa(fun(B,C),fun(A,C),aa(fun(A,C),fun(fun(B,C),fun(A,C)),aTP_Lamp_adu(set(B),fun(fun(A,C),fun(fun(B,C),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(C),C,complete_Inf_Inf(C),aa(set(B),set(C),image2(B,C,aa(A,fun(B,C),aa(fun(B,C),fun(A,fun(B,C)),aTP_Lamp_adt(fun(A,C),fun(fun(B,C),fun(A,fun(B,C))),Uua),Uub),Uuc)),Uu)) ) ).

% ATP.lambda_881
tff(fact_9065_ATP_Olambda__882,axiom,
    ! [A: $tType,C: $tType,D: $tType,B: $tType,Uu: set(B),Uua: fun(A,filter(C)),Uub: fun(B,filter(D)),Uuc: A] : aa(A,filter(product_prod(C,D)),aa(fun(B,filter(D)),fun(A,filter(product_prod(C,D))),aa(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D)))),aTP_Lamp_xk(set(B),fun(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,filter(product_prod(C,D))))),Uu),Uua),Uub),Uuc) = aa(set(filter(product_prod(C,D))),filter(product_prod(C,D)),complete_Inf_Inf(filter(product_prod(C,D))),aa(set(B),set(filter(product_prod(C,D))),image2(B,filter(product_prod(C,D)),aa(A,fun(B,filter(product_prod(C,D))),aa(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D)))),aTP_Lamp_xj(fun(A,filter(C)),fun(fun(B,filter(D)),fun(A,fun(B,filter(product_prod(C,D))))),Uua),Uub),Uuc)),Uu)) ).

% ATP.lambda_882
tff(fact_9066_ATP_Olambda__883,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(B,set(A)),Uua: fun(C,set(A)),Uub: set(C),Uuc: B] : aa(B,set(A),aa(set(C),fun(B,set(A)),aa(fun(C,set(A)),fun(set(C),fun(B,set(A))),aTP_Lamp_lp(fun(B,set(A)),fun(fun(C,set(A)),fun(set(C),fun(B,set(A)))),Uu),Uua),Uub),Uuc) = aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image2(C,set(A),aa(B,fun(C,set(A)),aa(fun(C,set(A)),fun(B,fun(C,set(A))),aTP_Lamp_lo(fun(B,set(A)),fun(fun(C,set(A)),fun(B,fun(C,set(A)))),Uu),Uua),Uuc)),Uub)) ).

% ATP.lambda_883
tff(fact_9067_ATP_Olambda__884,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_kk(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image2(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_kj(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub)) ) ).

% ATP.lambda_884
tff(fact_9068_ATP_Olambda__885,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_oe(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)))) ) ).

% ATP.lambda_885
tff(fact_9069_ATP_Olambda__886,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(A,fun(B,set(C))),Uub: B,Uuc: A] : aa(A,set(C),aa(B,fun(A,set(C)),aa(fun(A,fun(B,set(C))),fun(B,fun(A,set(C))),aTP_Lamp_aew(set(product_prod(B,B)),fun(fun(A,fun(B,set(C))),fun(B,fun(A,set(C)))),Uu),Uua),Uub),Uuc) = aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(B),set(set(C)),image2(B,set(C),aa(A,fun(B,set(C)),Uua,Uuc)),aa(set(B),set(B),image(B,B,Uu),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert2(B),Uub),bot_bot(set(B)))))) ).

% ATP.lambda_886
tff(fact_9070_ATP_Olambda__887,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_vd(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X: A] :
          ( ( member(A,X,Uu)
           => member(B,aa(A,B,Uuc,X),Uua) )
          & ( ~ member(A,X,Uu)
           => ( aa(A,B,Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_887
tff(fact_9071_ATP_Olambda__888,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(set(product_prod(C,B)),fun(A,fun(B,$o)),aTP_Lamp_ahd(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [Y4: C] :
          ( member(product_prod(A,C),aa(C,product_prod(A,C),product_Pair(A,C,Uub),Y4),Uu)
          & member(product_prod(C,B),aa(B,product_prod(C,B),product_Pair(C,B,Y4),Uuc),Uua) ) ) ).

% ATP.lambda_888
tff(fact_9072_ATP_Olambda__889,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( aa(C,$o,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_zq(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [X: A,Y4: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X),Y4) )
          & aa(A,$o,Uu,X)
          & aa(B,$o,Uua,Y4) ) ) ).

% ATP.lambda_889
tff(fact_9073_ATP_Olambda__890,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: A,Uub: A,Uuc: B,Uud: set(product_prod(C,B))] :
      aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(B,fun(set(product_prod(C,B)),set(product_prod(C,B))),aa(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))),aa(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B))))),aTP_Lamp_agq(C,fun(A,fun(A,fun(B,fun(set(product_prod(C,B)),set(product_prod(C,B)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(C,B)),set(product_prod(C,B)),aa(product_prod(C,B),fun(set(product_prod(C,B)),set(product_prod(C,B))),insert2(product_prod(C,B)),aa(B,product_prod(C,B),product_Pair(C,B,Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_890
tff(fact_9074_ATP_Olambda__891,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: A,Uua: B,Uub: B,Uuc: C,Uud: set(product_prod(A,C))] :
      aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(C,fun(set(product_prod(A,C)),set(product_prod(A,C))),aa(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))),aa(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C))))),aTP_Lamp_agx(A,fun(B,fun(B,fun(C,fun(set(product_prod(A,C)),set(product_prod(A,C)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(Uua = Uub,aa(set(product_prod(A,C)),set(product_prod(A,C)),aa(product_prod(A,C),fun(set(product_prod(A,C)),set(product_prod(A,C))),insert2(product_prod(A,C)),aa(C,product_prod(A,C),product_Pair(A,C,Uu),Uuc)),Uud),Uud) ).

% ATP.lambda_891
tff(fact_9075_ATP_Olambda__892,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_oa(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_892
tff(fact_9076_ATP_Olambda__893,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_on(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7311177749621191930dd_sum(A,C),aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_om(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_893
tff(fact_9077_ATP_Olambda__894,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_fh(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_fg(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_894
tff(fact_9078_ATP_Olambda__895,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_no(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(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_nn(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Uu),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),divide_divide(real,aa(nat,real,power_power(real,Uud),aa(nat,nat,minus_minus(nat,Uu),Uuc)),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uu),Uuc)))))) ).

% ATP.lambda_895
tff(fact_9079_ATP_Olambda__896,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_fk(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_fj(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,power_power(A,Uub),Uud)) ) ).

% ATP.lambda_896
tff(fact_9080_ATP_Olambda__897,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_aec(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uua),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uub),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uuc),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uud),bot_bot(set(A))))))),aa(set(product_prod(A,A)),set(A),field2(A),Uu))
        & ( ( ( Uua = Uuc )
            & ( Uub = Uud ) )
          | member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub)),bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud)),aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),Uu),id2(A)))
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Uua),Uuc),aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),Uu),id2(A))) )
          | ( ( bNF_We1388413361240627857o_max2(A,Uu,Uua,Uub) = bNF_We1388413361240627857o_max2(A,Uu,Uuc,Uud) )
            & ( Uua = Uuc )
            & member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Uub),Uud),aa(set(product_prod(A,A)),set(product_prod(A,A)),minus_minus(set(product_prod(A,A)),Uu),id2(A))) ) ) ) ) ).

% ATP.lambda_897
tff(fact_9081_ATP_Olambda__898,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_fg(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,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_898
tff(fact_9082_ATP_Olambda__899,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(C) )
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(A,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aa(B,fun(fun(A,C),fun(C,fun(A,$o))),aTP_Lamp_uc(fun(A,B),fun(B,fun(fun(A,C),fun(C,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V557655796197034286t_dist(C,aa(A,C,Uub,Uud),Uuc)),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uud),Uua)) ) ) ).

% ATP.lambda_899
tff(fact_9083_ATP_Olambda__900,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: A,Uub: A,Uuc: A,Uud: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aa(A,fun(A,fun(A,fun(A,$o))),aTP_Lamp_aht(set(product_prod(A,A)),fun(A,fun(A,fun(A,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( ( Uub = Uuc )
       => member(product_prod(A,A),aa(A,product_prod(A,A),product_Pair(A,A,Uua),Uud),Uu) ) ) ).

% ATP.lambda_900
tff(fact_9084_ATP_Olambda__901,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: int,Uud: B] : aa(B,A,aa(int,fun(B,A),aa(fun(B,A),fun(int,fun(B,A)),aa(B,fun(fun(B,A),fun(int,fun(B,A))),aTP_Lamp_yy(fun(B,A),fun(B,fun(fun(B,A),fun(int,fun(B,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uub,Uud)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Uuc)),power_int(A,aa(B,A,Uu,Uua),aa(int,int,minus_minus(int,Uuc),one_one(int))))) ) ).

% ATP.lambda_901
tff(fact_9085_ATP_Olambda__902,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_om(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),aa(set(A),C,aa(fun(A,C),fun(set(A),C),groups7121269368397514597t_prod(A,C),aa(B,fun(A,C),aTP_Lamp_ok(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),aa(set(A),set(A),minus_minus(set(A),Uu),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert2(A),Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_902
tff(fact_9086_ATP_Olambda__903,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_oc(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = divide_divide(B,aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_903
tff(fact_9087_ATP_Olambda__904,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_op(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)))),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_904
tff(fact_9088_ATP_Olambda__905,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_oj(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),divide_divide(B,aa(A,B,Uua,Uue),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_905
tff(fact_9089_ATP_Olambda__906,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_zj(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uue),Uuf)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uuf),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
        & ! [I4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uud))
           => ( ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I4)),X_13)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I4) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X8: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X8) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less(nat),Uue),X)
                    & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_906
tff(fact_9090_ATP_Olambda__907,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_mb($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_907
tff(fact_9091_ATP_Olambda__908,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_agz(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_908
tff(fact_9092_ATP_Olambda__909,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_aha(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_909
tff(fact_9093_ATP_Olambda__910,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_agc(set(B),fun(A,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_910
tff(fact_9094_ATP_Olambda__911,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_911
tff(fact_9095_ATP_Olambda__912,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_agu(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_912
tff(fact_9096_ATP_Olambda__913,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_ld(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_913
tff(fact_9097_ATP_Olambda__914,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_afz(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_914
tff(fact_9098_ATP_Olambda__915,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,$o),Uua: A] : aa(A,fun(B,$o),aTP_Lamp_xn(fun(B,$o),fun(A,fun(B,$o)),Uu),Uua) = Uu ).

% ATP.lambda_915
tff(fact_9099_ATP_Olambda__916,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,B),Uua: A] : aa(A,fun(B,B),aTP_Lamp_xp(fun(B,B),fun(A,fun(B,B)),Uu),Uua) = Uu ).

% ATP.lambda_916
tff(fact_9100_ATP_Olambda__917,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_jv(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_917
tff(fact_9101_ATP_Olambda__918,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_jt(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_918
tff(fact_9102_ATP_Olambda__919,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_nx(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_919
tff(fact_9103_ATP_Olambda__920,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ju(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_920
tff(fact_9104_ATP_Olambda__921,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_wt(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_921
tff(fact_9105_ATP_Olambda__922,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ke(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_922
tff(fact_9106_ATP_Olambda__923,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_lc(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_923
tff(fact_9107_ATP_Olambda__924,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cl(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_924
tff(fact_9108_ATP_Olambda__925,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mx(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_925
tff(fact_9109_ATP_Olambda__926,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_kp(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_926
tff(fact_9110_ATP_Olambda__927,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_pd(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_927
tff(fact_9111_ATP_Olambda__928,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_acw(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_928
tff(fact_9112_ATP_Olambda__929,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_kf(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_929
tff(fact_9113_ATP_Olambda__930,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aje(A,fun(A,$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_930
tff(fact_9114_ATP_Olambda__931,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_ej(complex,complex),Uu) = Uu ).

% ATP.lambda_931
tff(fact_9115_ATP_Olambda__932,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_dt(nat,nat),Uu) = Uu ).

% ATP.lambda_932
tff(fact_9116_ATP_Olambda__933,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_en(int,int),Uu) = Uu ).

% ATP.lambda_933
tff(fact_9117_ATP_Olambda__934,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rw(A,A),Uu) = Uu ) ).

% ATP.lambda_934
tff(fact_9118_ATP_Olambda__935,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rz(A,A),Uu) = Uu ) ).

% ATP.lambda_935
tff(fact_9119_ATP_Olambda__936,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_sb(A,A),Uu) = Uu ) ).

% ATP.lambda_936
tff(fact_9120_ATP_Olambda__937,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_jn(A,A),Uu) = Uu ) ).

% ATP.lambda_937
tff(fact_9121_ATP_Olambda__938,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_jq(A,A),Uu) = Uu ).

% ATP.lambda_938
tff(fact_9122_ATP_Olambda__939,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_yb(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_939
tff(fact_9123_ATP_Olambda__940,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,set(A),aTP_Lamp_ll(B,set(A)),Uu) = bot_bot(set(A)) ).

% ATP.lambda_940
tff(fact_9124_ATP_Olambda__941,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_jr(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_941
tff(fact_9125_ATP_Olambda__942,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_js(B,A),Uu) = bot_bot(A) ) ).

% ATP.lambda_942
tff(fact_9126_ATP_Olambda__943,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_yc(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_943
tff(fact_9127_ATP_Olambda__944,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,set(B),aTP_Lamp_aga(A,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_944
tff(fact_9128_ATP_Olambda__945,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bx(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_945
tff(fact_9129_ATP_Olambda__946,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_bt(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_946
tff(fact_9130_ATP_Olambda__947,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_bw(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_947
tff(fact_9131_ATP_Olambda__948,axiom,
    ! [B: $tType,A: $tType] :
      ( monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_acr(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_948
tff(fact_9132_ATP_Olambda__949,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ny(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_949
tff(fact_9133_ATP_Olambda__950,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: A] : aa(A,real,aTP_Lamp_ait(A,real),Uu) = zero_zero(real) ) ).

% ATP.lambda_950
tff(fact_9134_ATP_Olambda__951,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4867850818363320053vector(B)
        & real_V4867850818363320053vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_aja(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_951
tff(fact_9135_ATP_Olambda__952,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_at(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_952
tff(fact_9136_ATP_Olambda__953,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_akq(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_953
tff(fact_9137_ATP_Olambda__954,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ajy(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_954
tff(fact_9138_ATP_Olambda__955,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_aby(B,option(A)),Uu) = none(A) ).

% ATP.lambda_955
tff(fact_9139_ATP_Olambda__956,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_aca(A,option(B)),Uu) = none(B) ).

% ATP.lambda_956
tff(fact_9140_ATP_Olambda__957,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_ajw(A,B),Uu) = undefined(B) ).

% ATP.lambda_957
tff(fact_9141_ATP_Olambda__958,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_gi(real,$o),Uu)
    <=> $false ) ).

% ATP.lambda_958
tff(fact_9142_ATP_Olambda__959,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_sz(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_959
tff(fact_9143_ATP_Olambda__960,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_ao(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_960
tff(fact_9144_ATP_Olambda__961,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_afd(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_961
tff(fact_9145_ATP_Olambda__962,axiom,
    ! [A: $tType,Uu: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aTP_Lamp_abj(fun(A,$o),$o),Uu)
    <=> $true ) ).

% ATP.lambda_962
tff(fact_9146_ATP_Olambda__963,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_mf(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_963
tff(fact_9147_ATP_Olambda__964,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_mh(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_964

% Type constructors (830)
tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
    ! [A26: $tType,A27: $tType] :
      ( comple592849572758109894attice(A27)
     => counta4013691401010221786attice(fun(A26,A27)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A26: $tType,A27: $tType] :
      ( comple6319245703460814977attice(A27)
     => condit1219197933456340205attice(fun(A26,A27)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A26: $tType,A27: $tType] :
      ( counta3822494911875563373attice(A27)
     => counta3822494911875563373attice(fun(A26,A27)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A26: $tType,A27: $tType] :
      ( comple592849572758109894attice(A27)
     => comple592849572758109894attice(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A26: $tType,A27: $tType] :
      ( bounded_lattice(A27)
     => bounde4967611905675639751up_bot(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A26: $tType,A27: $tType] :
      ( bounded_lattice(A27)
     => bounde4346867609351753570nf_top(fun(A26,A27)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A26: $tType,A27: $tType] :
      ( comple6319245703460814977attice(A27)
     => comple6319245703460814977attice(fun(A26,A27)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A26: $tType,A27: $tType] :
      ( boolea8198339166811842893lgebra(A27)
     => boolea8198339166811842893lgebra(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__top,axiom,
    ! [A26: $tType,A27: $tType] :
      ( bounded_lattice(A27)
     => bounded_lattice_top(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
    ! [A26: $tType,A27: $tType] :
      ( bounded_lattice(A27)
     => bounded_lattice_bot(fun(A26,A27)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A26: $tType,A27: $tType] :
      ( comple6319245703460814977attice(A27)
     => comple9053668089753744459l_ccpo(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A26: $tType,A27: $tType] :
      ( semilattice_sup(A27)
     => semilattice_sup(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A26: $tType,A27: $tType] :
      ( semilattice_inf(A27)
     => semilattice_inf(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A26: $tType,A27: $tType] :
      ( distrib_lattice(A27)
     => distrib_lattice(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A26: $tType,A27: $tType] :
      ( bounded_lattice(A27)
     => bounded_lattice(fun(A26,A27)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A26: $tType,A27: $tType] :
      ( order_top(A27)
     => order_top(fun(A26,A27)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A26: $tType,A27: $tType] :
      ( order_bot(A27)
     => order_bot(fun(A26,A27)) ) ).

tff(tcon_fun___Countable_Ocountable,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( finite_finite(A26)
        & countable(A27) )
     => countable(fun(A26,A27)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A26: $tType,A27: $tType] :
      ( preorder(A27)
     => preorder(fun(A26,A27)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( finite_finite(A26)
        & finite_finite(A27) )
     => finite_finite(fun(A26,A27)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A26: $tType,A27: $tType] :
      ( lattice(A27)
     => lattice(fun(A26,A27)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A26: $tType,A27: $tType] :
      ( order(A27)
     => order(fun(A26,A27)) ) ).

tff(tcon_fun___Orderings_Otop,axiom,
    ! [A26: $tType,A27: $tType] :
      ( top(A27)
     => top(fun(A26,A27)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ord(A27)
     => ord(fun(A26,A27)) ) ).

tff(tcon_fun___Orderings_Obot,axiom,
    ! [A26: $tType,A27: $tType] :
      ( bot(A27)
     => bot(fun(A26,A27)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A26: $tType,A27: $tType] :
      ( uminus(A27)
     => uminus(fun(A26,A27)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A26: $tType,A27: $tType] :
      ( minus(A27)
     => minus(fun(A26,A27)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,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___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
    topolo8865339358273720382pology(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
    topolo5987344860129210374id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add(int) ).

tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
    topolo4211221413907600880p_mult(int) ).

tff(tcon_Int_Oint___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___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___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
    topological_t1_space(int) ).

tff(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord6961819062388156250ring_1(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__sup_2,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_4,axiom,
    distrib_lattice(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
    semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
    semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___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___Countable_Ocountable_5,axiom,
    countable(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___Orderings_Opreorder_6,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_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_7,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_8,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_9,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_10,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_11,axiom,
    minus(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_12,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_13,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_14,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_15,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_16,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_17,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_18,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_19,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_20,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_21,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_22,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_23,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_24,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_25,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_26,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_27,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_28,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_29,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_30,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_31,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_32,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_33,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_34,axiom,
    linord4140545234300271783up_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___Groups_Oordered__comm__monoid__add_42,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_44,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_45,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_46,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_47,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_48,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_51,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_52,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_53,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_54,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_55,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_56,axiom,
    distrib_lattice(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___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_74,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_75,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Countable_Ocountable_76,axiom,
    countable(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___Groups_Olinordered__ab__semigroup__add_117,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_118,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_119,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_120,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_121,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_122,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_123,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_124,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_125,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_126,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_127,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_128,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_129,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_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___Lattices_Odistrib__lattice_139,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_140,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_141,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_142,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_143,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_144,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_145,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_146,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_147,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_148,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_149,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_150,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_151,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_152,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_153,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_154,axiom,
    semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_155,axiom,
    zero_less_one(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___Countable_Ocountable_158,axiom,
    countable(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_159,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_160,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_161,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_162,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_163,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_164,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_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,
    ! [A26: $tType] : counta4013691401010221786attice(set(A26)) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_194,axiom,
    ! [A26: $tType] : condit1219197933456340205attice(set(A26)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_195,axiom,
    ! [A26: $tType] : counta3822494911875563373attice(set(A26)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_196,axiom,
    ! [A26: $tType] : comple592849572758109894attice(set(A26)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_197,axiom,
    ! [A26: $tType] : bounde4967611905675639751up_bot(set(A26)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_198,axiom,
    ! [A26: $tType] : bounde4346867609351753570nf_top(set(A26)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_199,axiom,
    ! [A26: $tType] : comple6319245703460814977attice(set(A26)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_200,axiom,
    ! [A26: $tType] : boolea8198339166811842893lgebra(set(A26)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__top_201,axiom,
    ! [A26: $tType] : bounded_lattice_top(set(A26)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice__bot_202,axiom,
    ! [A26: $tType] : bounded_lattice_bot(set(A26)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_203,axiom,
    ! [A26: $tType] : comple9053668089753744459l_ccpo(set(A26)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_204,axiom,
    ! [A26: $tType] : semilattice_sup(set(A26)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_205,axiom,
    ! [A26: $tType] : semilattice_inf(set(A26)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_206,axiom,
    ! [A26: $tType] : distrib_lattice(set(A26)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_207,axiom,
    ! [A26: $tType] : bounded_lattice(set(A26)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_208,axiom,
    ! [A26: $tType] : order_top(set(A26)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_209,axiom,
    ! [A26: $tType] : order_bot(set(A26)) ).

tff(tcon_Set_Oset___Countable_Ocountable_210,axiom,
    ! [A26: $tType] :
      ( finite_finite(A26)
     => countable(set(A26)) ) ).

tff(tcon_Set_Oset___Orderings_Opreorder_211,axiom,
    ! [A26: $tType] : preorder(set(A26)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_212,axiom,
    ! [A26: $tType] :
      ( finite_finite(A26)
     => finite_finite(set(A26)) ) ).

tff(tcon_Set_Oset___Lattices_Olattice_213,axiom,
    ! [A26: $tType] : lattice(set(A26)) ).

tff(tcon_Set_Oset___Orderings_Oorder_214,axiom,
    ! [A26: $tType] : order(set(A26)) ).

tff(tcon_Set_Oset___Orderings_Otop_215,axiom,
    ! [A26: $tType] : top(set(A26)) ).

tff(tcon_Set_Oset___Orderings_Oord_216,axiom,
    ! [A26: $tType] : ord(set(A26)) ).

tff(tcon_Set_Oset___Orderings_Obot_217,axiom,
    ! [A26: $tType] : bot(set(A26)) ).

tff(tcon_Set_Oset___Groups_Ouminus_218,axiom,
    ! [A26: $tType] : uminus(set(A26)) ).

tff(tcon_Set_Oset___Groups_Ominus_219,axiom,
    ! [A26: $tType] : minus(set(A26)) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_220,axiom,
    counta4013691401010221786attice($o) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_221,axiom,
    condit1219197933456340205attice($o) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_222,axiom,
    counta3822494911875563373attice($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_223,axiom,
    comple592849572758109894attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_224,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_225,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_226,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_227,axiom,
    bounde4967611905675639751up_bot($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_228,axiom,
    bounde4346867609351753570nf_top($o) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_229,axiom,
    comple6319245703460814977attice($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_230,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_231,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__top_232,axiom,
    bounded_lattice_top($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_233,axiom,
    bounded_lattice_bot($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_234,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_235,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_236,axiom,
    comple9053668089753744459l_ccpo($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_237,axiom,
    semilattice_sup($o) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_238,axiom,
    semilattice_inf($o) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_239,axiom,
    distrib_lattice($o) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_240,axiom,
    bounded_lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_241,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_242,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Countable_Ocountable_243,axiom,
    countable($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_244,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_245,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_246,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Lattices_Olattice_247,axiom,
    lattice($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_248,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Otop_249,axiom,
    top($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_250,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Orderings_Obot_251,axiom,
    bot($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_252,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_253,axiom,
    minus($o) ).

tff(tcon_List_Olist___Countable_Ocountable_254,axiom,
    ! [A26: $tType] :
      ( countable(A26)
     => countable(list(A26)) ) ).

tff(tcon_List_Olist___Nat_Osize_255,axiom,
    ! [A26: $tType] : size(list(A26)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_256,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_257,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_258,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
    condit5016429287641298734tinuum(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_259,axiom,
    ordere1937475149494474687imp_le(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
    topolo8458572112393995274pology(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom,
    topolo3112930676232923870pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
    real_V8999393235501362500lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
    real_V2822296259951069270ebra_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_260,axiom,
    semiri6575147826004484403cancel(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra,axiom,
    real_V4412858255891104859lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom,
    real_V5355595471888546746vector(real) ).

tff(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_261,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_262,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_263,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_264,axiom,
    linord2810124833399127020strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
    real_V822414075346904944vector(real) ).

tff(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_265,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_266,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_267,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_268,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_269,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_270,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
    real_V768167426530841204y_dist(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_271,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_272,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_273,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_274,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_275,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_276,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
    real_V8037385150606011577_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_277,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
    topolo7287701948861334536_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_278,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_279,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
    real_V7819770556892013058_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_280,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_281,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_282,axiom,
    archim2362893244070406136eiling(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
    real_V4867850818363320053vector(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_283,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_284,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_285,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_286,axiom,
    ring_15535105094025558882visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
    real_V7773925162809079976_field(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__monoid__add_287,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_288,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_289,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_290,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_291,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_292,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_293,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_294,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_295,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_296,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_297,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_298,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_299,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_300,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_301,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_302,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_303,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_304,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_305,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_306,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_307,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_308,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_309,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_310,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_311,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_312,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_313,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_314,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_315,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_316,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_317,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_318,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_319,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_320,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_321,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_322,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_323,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_324,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_325,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_326,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_327,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_328,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_329,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_330,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_331,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_332,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_333,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_334,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_335,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_336,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_337,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_338,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_339,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_340,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_341,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_342,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_343,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_344,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_345,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_346,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_347,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_348,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_349,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_350,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_351,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_352,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_353,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_354,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_355,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_356,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_357,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_358,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_359,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_360,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Countable_Ocountable_361,axiom,
    countable(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_362,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Nat_Osize_363,axiom,
    size(char) ).

tff(tcon_Sum__Type_Osum___Countable_Ocountable_364,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( countable(A26)
        & countable(A27) )
     => countable(sum_sum(A26,A27)) ) ).

tff(tcon_Sum__Type_Osum___Finite__Set_Ofinite_365,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( finite_finite(A26)
        & finite_finite(A27) )
     => finite_finite(sum_sum(A26,A27)) ) ).

tff(tcon_Sum__Type_Osum___Nat_Osize_366,axiom,
    ! [A26: $tType,A27: $tType] : size(sum_sum(A26,A27)) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_367,axiom,
    ! [A26: $tType] : condit1219197933456340205attice(filter(A26)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_368,axiom,
    ! [A26: $tType] : counta3822494911875563373attice(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_369,axiom,
    ! [A26: $tType] : bounde4967611905675639751up_bot(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_370,axiom,
    ! [A26: $tType] : bounde4346867609351753570nf_top(filter(A26)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_371,axiom,
    ! [A26: $tType] : comple6319245703460814977attice(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_372,axiom,
    ! [A26: $tType] : bounded_lattice_top(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_373,axiom,
    ! [A26: $tType] : bounded_lattice_bot(filter(A26)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_374,axiom,
    ! [A26: $tType] : comple9053668089753744459l_ccpo(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_375,axiom,
    ! [A26: $tType] : semilattice_sup(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_376,axiom,
    ! [A26: $tType] : semilattice_inf(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_377,axiom,
    ! [A26: $tType] : distrib_lattice(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_378,axiom,
    ! [A26: $tType] : bounded_lattice(filter(A26)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_379,axiom,
    ! [A26: $tType] : order_top(filter(A26)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_380,axiom,
    ! [A26: $tType] : order_bot(filter(A26)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_381,axiom,
    ! [A26: $tType] : preorder(filter(A26)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_382,axiom,
    ! [A26: $tType] : lattice(filter(A26)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_383,axiom,
    ! [A26: $tType] : order(filter(A26)) ).

tff(tcon_Filter_Ofilter___Orderings_Otop_384,axiom,
    ! [A26: $tType] : top(filter(A26)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_385,axiom,
    ! [A26: $tType] : ord(filter(A26)) ).

tff(tcon_Filter_Ofilter___Orderings_Obot_386,axiom,
    ! [A26: $tType] : bot(filter(A26)) ).

tff(tcon_Option_Ooption___Countable_Ocountable_387,axiom,
    ! [A26: $tType] :
      ( countable(A26)
     => countable(option(A26)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_388,axiom,
    ! [A26: $tType] :
      ( finite_finite(A26)
     => finite_finite(option(A26)) ) ).

tff(tcon_Option_Ooption___Nat_Osize_389,axiom,
    ! [A26: $tType] : size(option(A26)) ).

tff(tcon_String_Oliteral___Groups_Osemigroup__add_390,axiom,
    semigroup_add(literal) ).

tff(tcon_String_Oliteral___Countable_Ocountable_391,axiom,
    countable(literal) ).

tff(tcon_String_Oliteral___Orderings_Opreorder_392,axiom,
    preorder(literal) ).

tff(tcon_String_Oliteral___Orderings_Olinorder_393,axiom,
    linorder(literal) ).

tff(tcon_String_Oliteral___Groups_Omonoid__add_394,axiom,
    monoid_add(literal) ).

tff(tcon_String_Oliteral___Orderings_Oorder_395,axiom,
    order(literal) ).

tff(tcon_String_Oliteral___Orderings_Oord_396,axiom,
    ord(literal) ).

tff(tcon_String_Oliteral___Groups_Ozero_397,axiom,
    zero(literal) ).

tff(tcon_String_Oliteral___Groups_Oplus_398,axiom,
    plus(literal) ).

tff(tcon_String_Oliteral___Nat_Osize_399,axiom,
    size(literal) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_400,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_401,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_402,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_403,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_404,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_405,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_406,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_407,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_408,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_409,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_410,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_411,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_412,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_413,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_414,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_415,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_416,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_417,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_418,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_419,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_420,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_421,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_422,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_423,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_424,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_425,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_426,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_427,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_428,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_429,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_430,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_431,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_432,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_433,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_434,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_435,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_436,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_437,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_438,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_439,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_440,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_441,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_442,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_443,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_444,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_445,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_446,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_447,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_448,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_449,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_450,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_451,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_452,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_453,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_454,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_455,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_456,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_457,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_458,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_459,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_460,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_461,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_462,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_463,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_464,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_465,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_466,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_467,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_468,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_469,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_470,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_471,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_472,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_473,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_474,axiom,
    counta4013691401010221786attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_475,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_476,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_477,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_478,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_479,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_480,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_481,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_482,axiom,
    bounde4346867609351753570nf_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_483,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_484,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_485,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_486,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_487,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_488,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_489,axiom,
    bounded_lattice_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_490,axiom,
    bounded_lattice_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_491,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_492,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_493,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_494,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_495,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_496,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_497,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_498,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_499,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_500,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_501,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_502,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_503,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_504,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_505,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_506,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_507,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_508,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_509,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_510,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_511,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable_Ocountable_512,axiom,
    countable(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_513,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_514,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_515,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_516,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_517,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_518,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_519,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_520,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_521,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_522,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_523,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Otop_524,axiom,
    top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_525,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Obot_526,axiom,
    bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_527,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_528,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_529,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_530,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_531,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_532,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_533,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_534,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( topolo4958980785337419405_space(A26)
        & topolo4958980785337419405_space(A27) )
     => topolo4958980785337419405_space(product_prod(A26,A27)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_535,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( topological_t2_space(A26)
        & topological_t2_space(A27) )
     => topological_t2_space(product_prod(A26,A27)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_536,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( topological_t1_space(A26)
        & topological_t1_space(A27) )
     => topological_t1_space(product_prod(A26,A27)) ) ).

tff(tcon_Product__Type_Oprod___Countable_Ocountable_537,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( countable(A26)
        & countable(A27) )
     => countable(product_prod(A26,A27)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_538,axiom,
    ! [A26: $tType,A27: $tType] :
      ( ( finite_finite(A26)
        & finite_finite(A27) )
     => finite_finite(product_prod(A26,A27)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_539,axiom,
    ! [A26: $tType,A27: $tType] : size(product_prod(A26,A27)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_540,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_541,axiom,
    counta4013691401010221786attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_542,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_543,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_544,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_545,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_546,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_547,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_548,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_549,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_550,axiom,
    bounded_lattice_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_551,axiom,
    bounded_lattice_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_552,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_553,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_554,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_555,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_556,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_557,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_558,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_559,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable_Ocountable_560,axiom,
    countable(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_561,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_562,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_563,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_564,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_565,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Otop_566,axiom,
    top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_567,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Obot_568,axiom,
    bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_569,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_570,axiom,
    minus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_571,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_572,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_573,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_574,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_575,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_576,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_577,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_578,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_579,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_580,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_581,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_582,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_583,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_584,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_585,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_586,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_587,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_588,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_589,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_590,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_591,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_592,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_593,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_594,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_595,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_596,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_597,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_598,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_599,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_600,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_601,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_602,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_603,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_604,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_605,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_606,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_607,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_608,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_609,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_610,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_611,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_612,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_613,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_614,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_615,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_616,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_617,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_618,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_619,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_620,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_621,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_622,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_623,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_624,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_625,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_626,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_627,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_628,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_629,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_630,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_631,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_632,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_633,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_634,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_635,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_636,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_637,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_638,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_639,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_640,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_641,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_642,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_643,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_644,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_645,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_646,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_647,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_648,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_649,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_650,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_651,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_652,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_653,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_654,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_655,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_656,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_657,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_658,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_659,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_660,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_661,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_662,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_663,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_664,axiom,
    size(vEBT_VEBT) ).

% Helper facts (4)
tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P: fun(A,$o),X10: A] :
      ( ~ aa(fun(A,$o),$o,fAll(A),P)
      | aa(A,$o,P,X10) ) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X10: A,Y8: A] :
      ( ( X10 != Y8 )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y8) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X10: A,Y8: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X10),Y8)
      | ( X10 = Y8 ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X8: A] : aa(A,$o,P,X8) ) ) ).

% Free types (2)
tff(tfree_0,hypothesis,
    real_V4867850818363320053vector(a) ).

tff(tfree_1,hypothesis,
    semiring_1(a) ).

% Conjectures (1)
tff(conj_0,conjecture,
    member(nat,y,aa(set(nat),set(nat),aa(set(nat),fun(set(nat),set(nat)),sup_sup(set(nat)),vEBT_VEBT_set_vebt(t)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert2(nat),x),bot_bot(set(nat))))) ).

%------------------------------------------------------------------------------